home
Algebra
569 total questions
loading...
loading...
users
loading...
add
Show All Subdomains
Hide All Subdomains
Question 1 1 of 569 selected Linear Equations In 1 Variable H

12x+284-s13=r(x-8)

In the given equation, s and r are constants, and s>0. If the equation has infinitely many solutions, what is the value of s ?

Show Answer Correct Answer: 403

The correct answer is 403. For a linear equation in one variable to have infinitely many solutions, the coefficients of the variable must be equal on both sides of the equation and the constant terms must also be equal on both sides of the equation. The given equation can be rewritten as 4(3x+7)4-s13=r(x-8), or 3x+7-s13=r(x-8). Applying the distributive property to the right-hand side of this equation yields 3x+7-s13=rx-8r. For this equation to have infinitely many solutions, the coefficients of x must be equal, so it follows that 3=r. Additionally, the constant terms must be equal, which means 7-s13=-8r. Substituting 3 for r in this equation yields 7-s13=-8(3), or 7-s13=-24. Adding s13 to both sides of this equation yields 7=-24+s13. Adding 24 to both sides of this equation yields 31=s13. Multiplying both sides of this equation by 13 yields 403=s. Therefore, if the equation has infinitely many solutions, the value of s is 403.

Question 2 2 of 569 selected Systems Of 2 Linear Equations In 2 Variables H

  • For the first line in the system:
    • The line slants gradually down from left to right.
    • The line passes through the following points:
      • (3 comma 2)
      • (8 comma 0)
  • For the second line in the system:
    • The line slants gradually down from left to right.
    • The line passes through the following points:
      • (3 comma 4)
      • (8 comma 0)

What system of linear equations is represented by the lines shown?

  1. 8 x + 4 y = 32

    - 10 x - 4 y = -64

  2. 8 x - 4 y = 32

    - 10 x + 4 y = -64

  3. 4 x - 10 y = 32

    - 8 x + 10 y = -64

  4. 4 x + 10 y = 32

    - 8 x - 10 y = -64

Show Answer Correct Answer: D

Choice D is correct. A line in the xy-plane that passes through the points (x1,y1) and (x2,y2) has slope m , where m=y2-y1x2-x1, and can be defined by an equation of the form y-y1=m(x-x1). One of the lines shown in the graph passes through the points (8,0) and (3,4). Substituting 8 for x1, 0 for y1, 3 for x2, and 4 for y2 in the equation m=y2-y1x2-x1 yields m=4-03-8, or m=-45. Substituting - 4 5 for m , 8 for x1 and 0 for y1 in the equation y-y1=m(x-x1) yields y-0=-45(x-8), which is equivalent to y=-45x+325. Adding 45x to both sides of this equation yields 45x+y=325. Multiplying both sides of this equation by -10 yields - 8 x - 10 y = -64 . Therefore, an equation of this line is - 8 x - 10 y = -64 . Similarly, the other line shown in the graph passes through the points (8,0) and (3,2). Substituting 8 for x1, 0 for y1, 3 for x2, and 2 for y2 in the equation m=y2-y1x2-x1 yields m=2-03-8, or m = - 2 5 . Substituting - 2 5 for m , 8 for x1, and 0 for y1 in the equation y-y1=m(x-x1) yields y-0=-25(x-8), which is equivalent to y=-25x+165. Adding 25x to both sides of this equation yields 25x+y=165. Multiplying both sides of this equation by 10 yields 4 x + 10 y = 32 . Therefore, an equation of this line is 4 x + 10 y = 32 . So, the system of linear equations represented by the lines shown is 4 x + 10 y = 32 and - 8 x - 10 y = -64 .

Choice A is incorrect and may result from conceptual or calculation errors. 

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Question 3 3 of 569 selected Linear Functions E

The function f is defined by f(x)=25x+30. What is the value of f(x) when x = 2 ?

  1. 50

  2. 57

  3. 80

  4. 110

Show Answer Correct Answer: C

Choice C is correct. It’s given that the function f is defined by f(x)=25x+30. Substituting 2 for x in this equation yields f(2)=25(2)+30, which is equivalent to f(2)=50+30, or f(2)=80. Therefore, the value of f(x) is 80 when x=2.

Choice A is incorrect. This is the value of 25(2), not 25(2)+30.

Choice B is incorrect. This is the value of 25+2+30, not 25(2)+30.

Choice D is incorrect. This is the value of (25+30)(2), not 25(2)+30.

Question 4 4 of 569 selected Linear Equations In 2 Variables M

Line k is defined by y=-173x+5. Line j is perpendicular to line k in the xy-plane. What is the slope of line j ?

Show Answer Correct Answer: .1764, .1765, 3/17

The correct answer is 3 17 . It’s given that line j is perpendicular to line k in the xy-plane. This means that the slope of line j is the negative reciprocal of the slope of line k . The equation of line k , y=-173x+5, is written in slope-intercept form y = m x + b , where m is the slope of the line and b is the y-coordinate of the y-intercept of the line. It follows that the slope of line k is - 17 3 . The negative reciprocal of a number is -1 divided by the number. Therefore, the negative reciprocal of - 17 3 is -1-173, or 3 17 . Thus, the slope of line j is 3 17 . Note that 3/17, .1764, .1765, and 0.176 are examples of ways to enter a correct answer.

Question 5 5 of 569 selected Systems Of 2 Linear Equations In 2 Variables H

2(8x)+4(7y)=12

-2(8x)+4(7y)=12

The solution to the given system of equations is (x,y). What is the value of 8 x + 7 y ?

Show Answer Correct Answer: 3

The correct answer is 3 . Adding the second equation to the first equation in the given system of equations yields (2(8x)-2(8x))+(4(7y)+4(7y))=12+12, or 8(7y)=24. Dividing both sides of this equation by 8 yields 7 y = 3 . Substituting 3 for 7 y in the first equation, 2(8x)+4(7y)=12, yields 2(8x)+4(3)=12, or 2(8x)+12=12. Subtracting 12 from both sides of this equation yields 2(8x)=0. Dividing both sides of this equation by 2 yields 8 x = 0 . Substituting 0 for 8 x and 3 for 7 y in the expression 8x+7y yields 0+3, or 3 . Therefore, the value of 8 x + 7 y is 3 .

Question 6 6 of 569 selected Linear Inequalities In 1 Or 2 Variables M

A cargo helicopter delivers only 100-pound packages and 120-pound packages. For each delivery trip, the helicopter must carry at least 10 packages, and the total weight of the packages can be at most 1,100 pounds. What is the maximum number of 120-pound packages that the helicopter can carry per trip?

  1. 2

  2. 4

  3. 5

  4. 6

Show Answer Correct Answer: C

Choice C is correct. Let a equal the number of 120-pound packages, and let b equal the number of 100-pound packages. It’s given that the total weight of the packages can be at most 1,100 pounds: the inequality 120 a, plus 100 b, is less than or equal to 1,100 represents this situation. It’s also given that the helicopter must carry at least 10 packages: the inequality a, plus b, is greater than or equal to 10 represents this situation. Values of a and b that satisfy these two inequalities represent the allowable numbers of 120-pound packages and 100-pound packages the helicopter can transport. To maximize the number of 120-pound packages, a, in the helicopter, the number of 100-pound packages, b, in the helicopter needs to be minimized. Expressing b in terms of a in the second inequality yields b is greater than or equal to, 10 minus a, so the minimum value of b is equal to 10 minus a. Substituting 10 minus a for b in the first inequality results in 120 a, plus 100, times, open parenthesis, 10 minus a, close parenthesis, is less than or equal to 1,100. Using the distributive property to rewrite this inequality yields 120 a, plus 1,000, minus 100 a, is less than or equal to 1,100, or 20 a, plus 1,000, is less than or equal to 1,100. Subtracting 1,000 from both sides of this inequality yields 20 a, is less than or equal to 100. Dividing both sides of this inequality by 20 results in a, is less than or equal to 5. This means that the maximum number of 120-pound packages that the helicopter can carry per trip is 5.

Choices A, B, and D are incorrect and may result from incorrectly creating or solving the system of inequalities.

Question 7 7 of 569 selected Linear Equations In 1 Variable E

If 2 x + 3 = 9 , what is the value of 6 x - 1 ?

Show Answer Correct Answer: 17

The correct answer is 17 . It’s given that 2x+3=9. Multiplying each side of this equation by 3 yields 3(2x+3)=3(9), or 6x+9=27. Subtracting 10 from each side of this equation yields 6x+9-10=27-10, or 6x-1=17. Therefore, the value of 6x-1 is 17 .

Question 8 8 of 569 selected Linear Functions E

The function f is defined by f(x)=8x. For what value of x does f(x)=72?

  1. 8

  2. 9

  3. 64

  4. 80

Show Answer Correct Answer: B

Choice B is correct. Substituting 72 for f(x) in the given function yields 72=8x. Dividing each side of this equation by 8 yields 9=x. Therefore, f(x)=72 when the value of x is 9 .

Choice A is incorrect. This is the value of x for which f(x)=64, not f(x)=72.

Choice C is incorrect. This is the value of x for which f(x)=512, not f(x)=72.

Choice D is incorrect. This is the value of x for which f(x)=640, not f(x)=72.

Question 9 9 of 569 selected Linear Functions E

f(x)=4x+b

For the linear function f , b is a constant and f(7)=28. What is the value of b ?

  1. 0

  2. 1

  3. 4

  4. 7

Show Answer Correct Answer: A

Choice A is correct. For the linear function f , it’s given that f(7)=28. Substituting 7 for x and 28 for f(x) in the given function yields 28=4(7)+b, or 28=28+b. Subtracting 28 from each side of this equation yields 0=b. Therefore, the value of b is 0 .

Choice B is incorrect. Substituting 1 for b in the given function yields f(x)=4x+1. For this function, when the value of x is 7 , the value of f(x) is 29 , not 28 .

Choice C is incorrect. Substituting 4 for b in the given function yields f(x)=4x+4. For this function, when the value of x is 7 , the value of f(x) is 32 , not 28 .

Choice D is incorrect. Substituting 7 for b in the given function yields f(x)=4x+7. For this function, when the value of x is 7 , the value of f(x) is 35 , not 28 .

Question 10 10 of 569 selected Linear Equations In 2 Variables H
x y
k 13
k + 7 -15

The table gives the coordinates of two points on a line in the xy-plane. The y-intercept of the line is (k-5,b), where k and b are constants. What is the value of b ?

Show Answer Correct Answer: 33

The correct answer is 33 . It’s given in the table that the coordinates of two points on a line in the xy-plane are (k,13) and (k+7,-15). The y-intercept is another point on the line. The slope computed using any pair of points from the line will be the same. The slope of a line, m , between any two points, (x1,y1) and (x2,y2), on the line can be calculated using the slope formula, m=(y2-y1)(x2-x1). It follows that the slope of the line with the given points from the table, (k,13) and (k+7,-15), is m=-15-13k+7-k, which is equivalent to m=-287, or m=-4. It's given that the y-intercept of the line is (k-5,b). Substituting -4 for m and the coordinates of the points (k-5,b) and (k,13) into the slope formula yields -4=13-bk-(k-5), which is equivalent to -4=13-bk-k+5, or -4=13-b5. Multiplying both sides of this equation by 5 yields -20=13-b. Subtracting 13 from both sides of this equation yields -33=-b. Dividing both sides of this equation by -1 yields b=33. Therefore, the value of b is 33 .

Question 11 11 of 569 selected Linear Functions E

Gabriella deposits $35 in a savings account at the end of each week. At the beginning of the 1st week of a year there was $600 in that savings account. How much money, in dollars, will be in the account at the end of the 4th week of that year?

  1. 460

  2. 635

  3. 639

  4. 740

Show Answer Correct Answer: D

Choice D is correct. It’s given that at the beginning of the 1st week of the year there was $600 in a savings account and Gabriella deposits $35 in that savings account at the end of each week. Therefore, the amount of money, in dollars, in the savings account at the end of the 4th week of that year is 600+4(35), or 740 .

Choice A is incorrect. This is the amount of money, in dollars, that will be in the account at the end of the 4th week if Gabriella withdraws, rather than deposits, $35 at the end of each week.

Choice B is incorrect. This is the amount of money, in dollars, that will be in the account at the end of the 1st week, not the 4th week.

Choice C is incorrect and may result from conceptual or calculation errors.

Question 12 12 of 569 selected Systems Of 2 Linear Equations In 2 Variables H

3 x = 36 y - 45

One of the two equations in a system of linear equations is given. The system has no solution. Which equation could be the second equation in this system?

  1. x = 4 y

  2. 1 3 x = 4 y

  3. x = 12 y - 15

  4. 1 3 x = 12 y - 15

Show Answer Correct Answer: B

Choice B is correct. A system of two linear equations in two variables, x and y , has no solution when the lines in the xy-plane representing the equations are parallel and distinct. Two lines are parallel and distinct if their slopes are the same and their y-intercepts are different. The slope of the graph of the given equation, 3x=36y-45, in the xy-plane can be found by rewriting the equation in the form y=mx+b, where m is the slope of the graph and (0,b) is the y-intercept. Adding 45 to each side of the given equation yields 3x+45=36y. Dividing each side of this equation by 36 yields 112x+54=y, or y=112x+54. It follows that the slope of the graph of the given equation is 112 and the y-intercept is (0,54). Therefore, the graph of the second equation in the system must also have a slope of 112, but must not have a y-intercept of (0,54). Multiplying each side of the equation given in choice B by 14 yields 112x=y, or y=112x. It follows that the graph representing the equation in choice B has a slope of 112 and a y-intercept of (0,0). Since the slopes of the graphs of the two equations are equal and the y-intercepts of the graphs of the two equations are different, the equation in choice B could be the second equation in the system.

Choice A is incorrect. This equation can be rewritten as y=14x. It follows that the graph of this equation has a slope of 14, so the system consisting of this equation and the given equation has exactly one solution, rather than no solution.

Choice C is incorrect. This equation can be rewritten as y=112x+54. It follows that the graph of this equation has a slope of 112and a y-intercept of (0,54), so the system consisting of this equation and the given equation has infinitely many solutions, rather than no solution.

Choice D is incorrect. This equation can be rewritten as y=136x+54. It follows that the graph of this equation has a slope of 136, so the system consisting of this equation and the given equation has exactly one solution, rather than no solution.

Question 13 13 of 569 selected Systems Of 2 Linear Equations In 2 Variables H

negative x plus y, equals negative 3 point 5, and, x plus 3 y, equals 9 point 5

If the ordered pair x comma y satisfies the system of equations above, what is the value of y ?

Show Answer

The correct answer is three halves. One method for solving the system of equations for y is to add corresponding sides of the two equations. Adding the left-hand sides gives open parenthesis, negative x plus y, close parenthesis, plus, open parenthesis, x plus 3 y, close parenthesis, or 4y. Adding the right-hand sides yields negative 3 point 5 plus 9 point 5, equals 6. It follows that 4 y equals 6. Finally, dividing both sides of 4 y equals 6 by 4 yields y equals six fourths or three halves. Note that 3/2 and 1.5 are examples of ways to enter a correct answer.

 

Question 14 14 of 569 selected Systems Of 2 Linear Equations In 2 Variables M
one half y equals 4
x minus, one half y, equals 2

The system of equations above has solution (x, y). What is the value of x ?

  1. 3

  2. the fraction 7 over 2​​​​​​​

  3. 4

  4. 6

Show Answer Correct Answer: D

Choice D is correct. Adding the corresponding sides of the two equations eliminates y and yields x equals 6, as shown.

The equation one half y equals 4, added to the equation x minus one half y, equals 2, gives the equation x plus 0, equals 6

If (x, y) is a solution to the system, then (x, y) satisfies both equations in the system and any equation derived from them. Therefore, x equals 6.

Choices A, B, and C are incorrect and may be the result of errors when solving the system.

 

Question 15 15 of 569 selected Linear Functions E

A veterinarian recommends that each day a certain rabbit should eat 25 calories per pound of the rabbit’s weight, plus an additional 11 calories. Which equation represents this situation, where c is the total number of calories the veterinarian recommends the rabbit should eat each day if the rabbit’s weight is x pounds?

  1. c = 25 x

  2. c = 36 x

  3. c = 11 x + 25

  4. c = 25 x + 11

Show Answer Correct Answer: D

Choice D is correct. It’s given that a veterinarian recommends that each day the rabbit should eat 25 calories per pound of the rabbit’s weight, plus an additional 11 calories. If the rabbit’s weight is x pounds, then multiplying 25 calories per pound by the rabbit’s weight, x pounds, yields 25 x calories. Adding the additional 11 calories that the rabbit should eat each day yields 25 x + 11 calories. It’s given that c is the total number of calories the veterinarian recommends the rabbit should eat each day if the rabbit’s weight is x pounds. Therefore, this situation can be represented by the equation c = 25 x + 11 .

Choice A is incorrect. This equation represents a situation where a veterinarian recommends that each day the rabbit should eat 25 calories per pound of the rabbit’s weight.

Choice B is incorrect. This equation represents a situation where a veterinarian recommends that each day the rabbit should eat 25+11, or 36 , calories per pound of the rabbit’s weight.

Choice C is incorrect. This equation represents a situation where a veterinarian recommends that each day the rabbit should eat 11 calories per pound of the rabbit’s weight, plus an additional 25 calories.

Question 16 16 of 569 selected Linear Functions E

The total cost f(x), in dollars, to lease a car for 36 months from a particular car dealership is given by f(x)=36x+1,000, where x is the monthly payment, in dollars. What is the total cost to lease a car when the monthly payment is $400?

  1. $13,400

  2. $13,000

  3. $15,400

  4. $37,400

Show Answer Correct Answer: C

Choice C is correct. It's given that f(x) is the total cost, in dollars, to lease a car from this dealership with a monthly payment of x dollars. Therefore, the total cost, in dollars, to lease the car when the monthly payment is $400 is represented by the value of f(x) when x=400. Substituting 400 for x in the equation f(x)=36x+1,000 yields  f(400)=36(400)+1,000, or f(400)=15,400. Thus, when the monthly payment is $400, the total cost to lease a car is $15,400

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 17 17 of 569 selected Linear Equations In 2 Variables H

The graph of the equation a, x plus k y, equals 6 is a line in the xy-plane, where a and k are constants. If the line contains the points with coordinates negative 2 comma negative 6 and 0 comma negative 3, what is the value of k ?

  1. negative 2

  2. negative 1

  3. 2

  4. 3

Show Answer Correct Answer: A

Choice A is correct. The value of k can be found using the slope-intercept form of a linear equation, y equals, m x plus b, where m is the slope and b is the y-coordinate of the y-intercept. The equation a, x plus, k y equals 6 can be rewritten in the form y equals, the negative of the fraction a, x, over k, end fraction, plus, the fraction 6 over k. One of the given points, with coordinates 0 comma negative 3, is the y-intercept. Thus, the y-coordinate of the y-intercept negative 3 must be equal to the fraction 6 over k. Multiplying both sides by k gives negative 3 k, equals 6. Dividing both sides by negative 3 gives k equals negative 2.

Choices B, C, and D are incorrect and may result from errors made rewriting the given equation.

 

Question 18 18 of 569 selected Linear Equations In 2 Variables H

Line l is defined by 3y+12x=5. Line n is perpendicular to line l in the xy-plane. What is the slope of line n ?

Show Answer Correct Answer: 0.25, 1/4

The correct answer is 14. For an equation in slope-intercept form y=mx+b, m represents the slope of the line in the xy-plane defined by this equation. It's given that line l is defined by 3y+12x=5. Subtracting 12x from both sides of this equation yields 3y=-12x+5. Dividing both sides of this equation by 3 yields y=-123x+53, or y=-4x+53. Thus, the slope of line l in the xy-plane is - 4 . Since line n is perpendicular to line l in the xy-plane, the slope of line n is the negative reciprocal of the slope of line l. The negative reciprocal of - 4 is -1(-4)=14. Note that 1/4 and .25 are examples of ways to enter a correct answer.

Question 19 19 of 569 selected Systems Of 2 Linear Equations In 2 Variables H

32y-14x=23-32y

12x+32=py+92

In the given system of equations, p is a constant. If the system has no solution, what is the value of p ?

Show Answer Correct Answer: 6

The correct answer is 6 . A system of two linear equations in two variables, x and y , has no solution if the lines represented by the equations in the xy-plane are parallel and distinct. Lines represented by equations in standard form, Ax+By=C and Dx+Ey=F, are parallel if the coefficients for x and y in one equation are proportional to the corresponding coefficients in the other equation, meaning DA=EB; and the lines are distinct if the constants are not proportional, meaning FC is not equal to DA or EB. The first equation in the given system is 32y-14x=23-32y. Multiplying each side of this equation by 12 yields 18y-3x=8-18y. Adding 18 y to each side of this equation yields 36y-3x=8, or -3x+36y=8. The second equation in the given system is 12x+32=py+92. Multiplying each side of this equation by 2 yields x+3=2py+9. Subtracting 2 p y from each side of this equation yields x+3-2py=9. Subtracting 3 from each side of this equation yields x-2py=6. Therefore, the two equations in the given system, written in standard form, are -3x+36y=8 and x-2py=6. As previously stated, if this system has no solution, the lines represented by the equations in the xy-plane are parallel and distinct, meaning the proportion 1-3=-2p36, or -13=-p18, is true and the proportion 68=1-3 is not true. The proportion 68=1-3 is not true. Multiplying each side of the true proportion, -13=-p18,  by -18 yields 6=p. Therefore, if the system has no solution, then the value of p is 6

Question 20 20 of 569 selected Linear Functions H

A window repair specialist charges $220 for the first two hours of repair plus an hourly fee for each additional hour. The total cost for 5 hours of repair is $400. Which function f gives the total cost, in dollars, for x hours of repair, where x2?

  1. f(x)=60x+100

  2. f(x)=60x+220

  3. f(x)=80x

  4. f(x)=80x+220

Show Answer Correct Answer: A

Choice A is correct. It’s given that the window repair specialist charges $220 for the first two hours of repair plus an hourly fee for each additional hour. Let n represent the hourly fee for each additional hour after the first two hours. Since it’s given that x is the number of hours of repair, it follows that the charge generated by the hourly fee after the first two hours can be represented by the expression n ( x - 2 ) . Therefore, the total cost, in dollars, for x hours of repair is  f(x)=220+n(x-2). It’s given that the total cost for 5 hours of repair is $400. Substituting 5 for x and 400 for f(x) into the equation  f(x)=220+n(x-2) yields 400=220+n(5-2), or 400=220+3n. Subtracting 220 from both sides of this equation yields 180 = 3 n . Dividing both sides of this equation by 3 yields n = 60 . Substituting 60 for n in the equation  f(x)=220+n(x-2) yields  f(x)=220+60(x-2), which is equivalent to  f(x)=220+60x-120, or  f(x)=60x+100. Therefore, the total cost, in dollars, for x hours of repair is  f(x)=60x+100.

Choice B is incorrect. This function represents the total cost, in dollars, for x hours of repair where the specialist charges $340, rather than $220, for the first two hours of repair.

Choice C is incorrect. This function represents the total cost, in dollars, for x hours of repair where the specialist charges $160, rather than $220, for the first two hours of repair, and an hourly fee of $80, rather than $60, after the first two hours.

Choice D is incorrect. This function represents the total cost, in dollars, for x hours of repair where the specialist charges $380, rather than $220, for the first two hours of repair, and an hourly fee of $80, rather than $60, after the first two hours.

Question 21 21 of 569 selected Linear Equations In 1 Variable H

Hector used a tool called an auger to remove corn from a storage bin at a constant rate. The bin contained 24,000 bushels of corn when Hector began to use the auger. After 5 hours of using the auger, 19,350 bushels of corn remained in the bin. If the auger continues to remove corn at this rate, what is the total number of hours Hector will have been using the auger when 12,840 bushels of corn remain in the bin?

  1.   3

  2.   7

  3.   8

  4. 12

Show Answer Correct Answer: D

Choice D is correct. After using the auger for 5 hours, Hector had removed 24,000 – 19,350 = 4,650 bushels of corn from the storage bin. During the 5-hour period, the auger removed corn from the bin at a constant rate of the fraction 4,650 over 5, equals 930 bushels per hour. Assuming the auger continues to remove corn at this rate, after x hours it will have removed 930x bushels of corn. Because the bin contained 24,000 bushels of corn when Hector started using the auger, the equation 24,000 – 930x = 12,840 can be used to find the number of hours, x, Hector will have been using the auger when 12,840 bushels of corn remain in the bin. Subtracting 12,840 from both sides of this equation and adding 930x to both sides of the equation yields 11,160 = 930x. Dividing both sides of this equation by 930 yields x = 12. Therefore, Hector will have been using the auger for 12 hours when 12,840 bushels of corn remain in the storage bin.

Choice A is incorrect. Three hours after Hector began using the auger, 24,000 – 3(930) = 21,210 bushels of corn remained, not 12,840. Choice B is incorrect. Seven hours after Hector began using the auger, 24,000 – 7(930) = 17,490 bushels of corn will remain, not 12,840. Choice C is incorrect. Eight hours after Hector began using the auger, 24,000 – 8(930) = 16,560 bushels of corn will remain, not 12,840.

Question 22 22 of 569 selected Linear Functions M

The function h is defined by h(x)=4x+28. The graph of y=h(x) in the xy-plane has an x-intercept at (a,0) and a y-intercept at (0,b), where a and b are constants. What is the value of a + b ?

  1. 21

  2. 28

  3. 32

  4. 35

Show Answer Correct Answer: A

Choice A is correct. The x-intercept of a graph in the xy-plane is the point on the graph where y = 0 . It's given that function h is defined by h(x)=4x+28. Therefore, the equation representing the graph of y=h(x) is y = 4 x + 28 . Substituting 0 for y in the equation y = 4 x + 28 yields 0 = 4 x + 28 . Subtracting 28 from both sides of this equation yields -28 = 4 x . Dividing both sides of this equation by 4 yields -7 = x . Therefore, the x-intercept of the graph of y=h(x) in the xy-plane is (-7,0). It's given that the x-intercept of the graph of y=h(x) is (a,0). Therefore, a = -7 . The y-intercept of a graph in the xy-plane is the point on the graph where x = 0 . Substituting 0 for x in the equation y = 4 x + 28 yields y=4(0)+28, or y = 28 . Therefore, the y-intercept of the graph of y=h(x) in the xy-plane is (0,28). It's given that the y-intercept of the graph of y=h(x) is (0,b). Therefore, b = 28 . If a = -7 and b = 28 , then the value of a + b is -7+28, or 21 .

Choice B is incorrect. This is the value of b , not a + b .

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect. This is the value of - a + b , not a + b .

Question 23 23 of 569 selected Linear Equations In 1 Variable E

If 4 x - 28 = -24 , what is the value of x - 7 ?

  1. -24

  2. -22

  3. -6

  4. -1

Show Answer Correct Answer: C

Choice C is correct. Dividing all terms in the given equation by 4 yields 4x4-284=-244, or x-7=-6. Therefore, the value of x - 7 is -6 .

Choice A is incorrect. This is the value of 4 x - 28 , not x - 7 .

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 24 24 of 569 selected Linear Equations In 2 Variables H
x y
18 130
23 160
26 178

For line h , the table shows three values of x and their corresponding values of y . Line k is the result of translating line h down 5 units in the xy-plane. What is the x-intercept of line k ?

  1. (-263,0)

  2. (-92,0)

  3. (-113,0)

  4. (-176,0)

Show Answer Correct Answer: D

Choice D is correct. The equation of line h can be written in slope-intercept form y = m x + b , where m is the slope of the line and (0,b) is the y-intercept of the line. It’s given that line h contains the points (18,130), (23,160), and (26,178). Therefore, its slope m can be found as 160-13023-18, or 6 . Substituting 6 for m in the equation y = m x + b yields y = 6 x + b . Substituting 130 for y and 18 for x in this equation yields 130=6(18)+b, or 130=108+b. Subtracting 108 from both sides of this equation yields 22 = b . Substituting 22 for b in y = 6 x + b yields y = 6 x + 22 . Since line k is the result of translating line h down 5 units, an equation of line k is y=6x+22-5, or y = 6 x + 17 . Substituting 0 for y in this equation yields 0 = 6 x + 17 . Solving this equation for x yields x=-176. Therefore, the x-intercept of line k is (-176,0).

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Question 25 25 of 569 selected Linear Functions H

An economist modeled the demand Q for a certain product as a linear function of the selling price P. The demand was 20,000 units when the selling price was $40 per unit, and the demand was 15,000 units when the selling price was $60 per unit. Based on the model, what is the demand, in units, when the selling price is $55 per unit?

  1. 16,250

  2. 16,500

  3. 16,750

  4. 17,500

Show Answer Correct Answer: A

Choice A is correct. Let the economist’s model be the linear function Q equals, m P plus b, where Q is the demand, P is the selling price, m is the slope of the line, and b is the y-coordinate of the y-intercept of the line in the xy-plane, where y equals Q. Two pairs of the selling price P and the demand Q are given. Using the coordinate pairs P comma Q, two points that satisfy the function are 40 comma 20,000 and 60 comma 15,000. The slope m of the function can be found using the formula m equals, the fraction with numerator Q sub 2, minus Q sub 1, and denominator P sub 2, minus P sub 1, end fraction. Substituting the given values into this formula yields m equals, the fraction with numerator 15,000 minus 20,000, and denominator 60 minus 40, end fraction, or  m equals, negative 250. Therefore, Q equals, negative 250 P plus b. The value of b can be found by substituting one of the points into the function. Substituting the values of P and Q from the point with coordinates 40 comma 20,000 yields 20,000 equals, negative 250 times 40, plus b, or 20,000 equals, negative 10,000 plus b. Adding 10,000 to both sides of this equation yields b equals 30,000 . Therefore, the linear function the economist used as the model is Q equals, negative 250 P plus 30,000. Substituting 55 for P yields Q equals, negative 250 times 55, plus 30,000, equals 16,250. It follows that when the selling price is $55 per unit, the demand is 16,250 units.

Choices B, C, and D are incorrect and may result from calculation or conceptual errors.

 

Question 26 26 of 569 selected Linear Functions H
x y
-12 -45
6 45

The table shows two values of x and their corresponding values of y . The graph of the linear equation representing this relationship passes through the point (14, a). What is the value of a ?

Show Answer Correct Answer: 16.25, 65/4

The correct answer is 654. The linear relationship between x and y can be represented by the equation y=mx+b, where m and b are constants. It's given in the table that when x=-12, y=-45. Substituting -12 for x and -45 for y in the equation y=mx+b yields -45=-12m+b, which can be rewritten as -45+12m=b. It's also given in the table that when x=6, y=45. Substituting 6 for x and 45 for y in the equation y=mx+b yields 45=6m+b, which can be rewritten as 45-6m=b. Substituting -45+12m for b in this equation yields 45-6m=-45+12m. Adding 6m to both sides of this equation yields 45=-45+18m. Adding 45 to both sides of this equation yields 90=18m. Dividing both sides of this equation by 18 yields 5=m, or m=5. Substituting 5 for m, -12 for x, and -45 for y in the equation y=mx+b yields -45=5(-12)+b, or -45=-60+b. Adding 60 to both sides of this equation yields 15=b. Therefore, m=5 and b=15. Substituting 5 for m and 15 for b in the equation y=mx+b yields y=5x+15. Thus, the equation y=5x+15 represents the linear relationship between x and y. It’s also given that the graph of the linear equation representing this relationship passes through the point (14,a). Substituting 14 for x and a for y in the equation y=5x+15 yields a=5(14)+15, which is equivalent to a=54+15, or a=654. Note that 65/4 and 16.25 are examples of ways to enter a correct answer.

Question 27 27 of 569 selected Linear Equations In 2 Variables H

A certain apprentice has enrolled in 85 hours of training courses. The equation 10 x + 15 y = 85 represents this situation, where x is the number of on-site training courses and y is the number of online training courses this apprentice has enrolled in. How many more hours does each online training course take than each on-site training course?

Show Answer Correct Answer: 5

The correct answer is 5 . It's given that the equation 10 x + 15 y = 85 represents the situation, where x is the number of on-site training courses, y is the number of online training courses, and 85 is the total number of hours of training courses the apprentice has enrolled in. Therefore, 10 x represents the number of hours the apprentice has enrolled in on-site training courses, and 15 y represents the number of hours the apprentice has enrolled in online training courses. Since x is the number of on-site training courses and y is the number of online training courses the apprentice has enrolled in, 10 is the number of hours each on-site course takes and 15 is the number of hours each online course takes. Subtracting these numbers gives 15-10, or 5 more hours each online training course takes than each on-site training course.

Question 28 28 of 569 selected Systems Of 2 Linear Equations In 2 Variables E

Hiro and Sofia purchased shirts and pants from a store. The price of each shirt purchased was the same and the price of each pair of pants purchased was the same. Hiro purchased 4 shirts and 2 pairs of pants for $86, and Sofia purchased 3 shirts and 5 pairs of pants for $166. Which of the following systems of linear equations represents the situation, if x represents the price, in dollars, of each shirt and y represents the price, in dollars, of each pair of pants?

  1. 4 x plus 2 y, equals 86, 
and, 
3 x plus 5 y, equals 166

  2. 4 x plus 3 y, equals 86, 
and, 
2 x plus 5 y, equals 166

  3. 4 x plus 2 y, equals 166, 
and, 
3 x plus 5 y, equals 86

  4. 4 x plus 3 y, equals 166, 
and, 
2 x plus 5 y, equals 86

Show Answer Correct Answer: A

Choice A is correct. Hiro purchased 4 shirts and each shirt cost x dollars, so he spent a total of 4x dollars on shirts. Likewise, Hiro purchased 2 pairs of pants, and each pair of pants cost y dollars, so he spent a total of 2y dollars on pants. Therefore, the total amount that Hiro spent was 4x + 2y. Since Hiro spent $86 in total, this can be modeled by the equation 4x + 2y = 86. Using the same reasoning, Sofia bought 3 shirts at x dollars each and 5 pairs of pants at y dollars each, so she spent a total of 3x + 5y dollars on shirts and pants. Since Sofia spent $166 in total, this can be modeled by the equation 3x + 5y = 166.

Choice B is incorrect and may be the result of switching the number of shirts Sofia purchased with the number of pairs of pants Hiro purchased. Choice C is incorrect and may be the result of switching the total price each person paid. Choice D is incorrect and may be the result of switching the total price each person paid as well as switching the number of shirts Sofia purchased with the number of pairs of pants Hiro purchased.

 

Question 29 29 of 569 selected Linear Functions E

d=16-x30

The equation shown gives the estimated amount of diesel d , in gallons, that remains in the gas tank of a truck after being driven x miles, where 0x480. What is the estimated amount of diesel, in gallons, that remains in the gas tank of the truck when x = 300 ?

  1. 0

  2. 6

  3. 14

  4. 16

Show Answer Correct Answer: B

Choice B is correct. It’s given that the equation d=16-x30 gives the estimated amount of diesel d , in gallons, that remains in the gas tank of the truck after being driven x miles. Substituting 300 for x in the given equation yields d=16-30030, which is equivalent to d=16-10, or d = 6 . Therefore, the estimated amount of diesel that remains in the gas tank of the truck when x = 300 is 6 gallons.

Choice A is incorrect. This is the estimated amount of diesel, in gallons, that will remain in the gas tank of the truck when x = 480 , not when x = 300 .

Choice C is incorrect. This is the estimated amount of diesel, in gallons, that will remain in the gas tank of the truck when x = 60 , not when x = 300 .

Choice D is incorrect. This is the estimated amount of diesel, in gallons, that will remain in the gas tank of the truck when x = 0 , not when x = 300 .

Question 30 30 of 569 selected Systems Of 2 Linear Equations In 2 Variables H

a x + b y = 72

6 x + 2 b y = 56

In the given system of equations, a and b are constants. The graphs of these equations in the xy-plane intersect at the point (4,y). What is the value of a ?

  1. 3

  2. 4

  3. 6

  4. 14

Show Answer Correct Answer: D

Choice D is correct. It’s given that the graphs of the given system of equations intersect at the point (4,y). Therefore, (4,y) is the solution to the given system. Multiplying the first equation in the given system by - 2 yields - 2 a x - 2 b y = - 144 . Adding this equation to the second equation in the system yields (-2a+6)x+(-2b+2b)y=(-144+56), or (-2a+6)x=-88. Since (4,y) is the solution to the system, the value of a can be found by substituting 4 for x in this equation, which yields (-2a+6)(4)=-88. Dividing both sides of this equation by 4 yields - 2 a + 6 = - 22 . Subtracting 6 from both sides of this equation yields - 2 a = - 28 . Dividing both sides of this equation by - 2 yields a = 14 .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Question 31 31 of 569 selected Linear Equations In 2 Variables E

A store sells two different-sized containers of a certain Greek yogurt. The store’s sales of this Greek yogurt totaled 1,277.94 dollars last month. The equation 5.48 x + 7.30 y = 1,277.94 represents this situation, where x is the number of smaller containers sold and y is the number of larger containers sold. According to the equation, which of the following represents the price, in dollars, of each smaller container?

  1. 5.48

  2. 7.30 y

  3. 7.30

  4. 5.48 x

Show Answer Correct Answer: A

Choice A is correct. It's given that the store's sales of a certain Greek yogurt totaled 1,277.94 dollars last month. It's also given that the equation 5.48x+7.30y=1,277.94 represents this situation, where x is the number of smaller containers sold and y is the number of larger containers sold. Since x represents the number of smaller containers of yogurt sold, the expression 5.48 x represents the total sales, in dollars, from smaller containers of yogurt. This means that x smaller containers of yogurt were sold at a price of 5.48 dollars each. Therefore, according to the equation, 5.48 represents the price, in dollars, of each smaller container.

Choice B is incorrect. This expression represents the total sales, in dollars, from selling y larger containers of yogurt.

Choice C is incorrect. This value represents the price, in dollars, of each larger container of yogurt.

Choice D is incorrect. This expression represents the total sales, in dollars, from selling x smaller containers of yogurt.

Question 32 32 of 569 selected Linear Equations In 1 Variable E

7(2x-3)=63

Which equation has the same solution as the given equation?

  1. 2 x - 3 = 9

  2. 2 x - 3 = 56

  3. 2 x - 21 = 63

  4. 2 x - 21 = 70

Show Answer Correct Answer: A

Choice A is correct. Dividing each side of the given equation by 7 yields 7(2x-3)7=637, or 2x-3=9. Therefore, the equation 2x-3=9 is equivalent to the given equation and has the same solution.

Choice B is incorrect. This equation is equivalent to 7(2x-3)=392, not 7(2x-3)=63.

Choice C is incorrect. Distributing 7 on the left-hand side of the given equation yields 14x-21=63, not 2x-21=63.

Choice D is incorrect. Distributing 7 on the left-hand side of the given equation yields 14x-21=63, not 2x-21=70.

Question 33 33 of 569 selected Linear Equations In 2 Variables E

Line k is defined by y = 3 x + 15 . Line j is perpendicular to line k in the xy-plane. What is the slope of line j ?

  1. - 1 3

  2. - 1 12

  3. - 1 18

  4. - 1 45

Show Answer Correct Answer: A

Choice A is correct. It's given that line j is perpendicular to line k in the xy-plane. It follows that the slope of line j is the opposite reciprocal of the slope of line k . The equation for line k is written in slope-intercept form y = m x + b , where m is the slope of the line and b is the y-coordinate of the y-intercept of the line. It follows that the slope of line k is 3 . The opposite reciprocal of a number is -1 divided by the number. Thus, the opposite reciprocal of 3 is -13. Therefore, the slope of line j is - 1 3 .

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 34 34 of 569 selected Linear Inequalities In 1 Or 2 Variables E

The point (8,2) in the xy-plane is a solution to which of the following systems of inequalities?

  1. x>0

    y>0

  2. x>0

    y<0

  3. x<0

    y>0

  4. x<0

    y<0

Show Answer Correct Answer: A

Choice A is correct. The given point, (8,2), is located in the first quadrant in the xy-plane. The system of inequalities in choice A represents all the points in the first quadrant in the xy-plane. Therefore, (8,2) is a solution to the system of inequalities in choice A.

Alternate approach: Substituting 8 for x in the first inequality in choice A, x>0, yields 8>0, which is true. Substituting 2 for y in the second inequality in choice A, y>0, yields 2>0, which is true. Since the coordinates of the point (8,2) make the inequalities x>0 and y>0 true, the point (8,2) is a solution to the system of inequalities consisting of x>0 and y>0.

Choice B is incorrect. This system of inequalities represents all the points in the fourth quadrant, not the first quadrant, in the xy-plane.

Choice C is incorrect. This system of inequalities represents all the points in the second quadrant, not the first quadrant, in the xy-plane.

Choice D is incorrect. This system of inequalities represents all the points in the third quadrant, not the first quadrant, in the xy-plane.

Question 35 35 of 569 selected Systems Of 2 Linear Equations In 2 Variables E

5 x = 15

- 4 x + y = -2

The solution to the given system of equations is (x,y). What is the value of x + y ?

  1. -17

  2. -13

  3. 13

  4. 17

Show Answer Correct Answer: C

Choice C is correct. Adding the second equation of the given system to the first equation yields 5x+(-4x+y)=15+(-2), which is equivalent to x+y=13. So the value of x+y is 13 .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect. This is the value of -(x+y).

Choice D is incorrect and may result from conceptual or calculation errors.

Question 36 36 of 569 selected Linear Functions H

The cost of renting a backhoe for up to 10 days is $270 for the first day and $135 for each additional day. Which of the following equations gives the cost y , in dollars, of renting the backhoe for x days, where x is a positive integer and x10?

  1. y = 270 x - 135

  2. y = 270 x + 135

  3. y = 135 x + 270

  4. y = 135 x + 135

Show Answer Correct Answer: D

Choice D is correct. It's given that the cost of renting a backhoe for up to 10 days is $270 for the first day and $135 for each additional day. Therefore, the cost y , in dollars, for x days, where x10, is the sum of the cost for the first day, $270 , and the cost for the additional x - 1 days, $135(x-1). It follows that y=270+135(x-1), which is equivalent to y=270+135x-135, or y=135x+135.

Choice A is incorrect. This equation represents a situation where the cost of renting a backhoe is $135 for the first day and $270 for each additional day.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Question 37 37 of 569 selected Linear Equations In 1 Variable E

What value of p satisfies the equation 5 p + 180 = 250 ?

  1. 14

  2. 65

  3. 86

  4. 250

Show Answer Correct Answer: A

Choice A is correct. Subtracting 180 from both sides of the given equation yields 5 p = 70 . Dividing both sides of this equation by 5 yields p = 14 . Therefore, the value of p that satisfies the equation 5p+180=250 is 14 .

Choice B is incorrect. This value of p satisfies the equation 5p+180=505.

Choice C is incorrect. This value of p satisfies the equation 5p+180=610.

Choice D is incorrect. This value of p satisfies the equation 5p+180=1,430.

Question 38 38 of 569 selected Linear Functions E

The front of a roller-coaster car is at the bottom of a hill and is 15 feet above the ground. If the front of the roller-coaster car rises at a constant rate of 8 feet per second, which of the following equations gives the height h, in feet, of the front of the roller-coaster car s seconds after it starts up the hill?

  1. h equals, 8 s, plus 15

  2. h equals, 15 s, plus the fraction 335 over 8

  3. h equals, 8 s, plus the fraction 335 over 15

  4. h equals, 15 s, plus 8

Show Answer Correct Answer: A

Choice A is correct. It’s given that the front of the roller-coaster car starts rising when it’s 15 feet above the ground. This initial height of 15 feet can be represented by a constant term, 15, in an equation. Each second, the front of the roller-coaster car rises 8 feet, which can be represented by 8s. Thus, the equation h equals, 8 s plus 15 gives the height, in feet, of the front of the roller-coaster car s seconds after it starts up the hill.

Choices B and C are incorrect and may result from conceptual errors in creating a linear equation. Choice D is incorrect and may result from switching the rate at which the roller-coaster car rises with its initial height.

 

Question 39 39 of 569 selected Linear Functions M

According to a model, the head width, in millimeters, of a worker bumblebee can be estimated by adding 0.6 to four times the body weight of the bee, in grams. According to the model, what would be the head width, in millimeters, of a worker bumblebee that has a body weight of 0.5 grams?

Show Answer

The correct answer is 2.6. According to the model, the head width, in millimeters, of a worker bumblebee can be estimated by adding 0.6 to 4 times the body weight, in grams, of the bee. Let x represent the body weight, in grams, of a worker bumblebee and let y represent the head width, in millimeters. Translating the verbal description of the model into an equation yields y equals, 0 point 6 plus 4 x. Substituting 0.5 grams for x in this equation yields y equals, 0 point 6, plus 4 times 0 point 5, or y equals 2 point 6. Therefore, a worker bumblebee with a body weight of 0.5 grams has an estimated head width of 2.6 millimeters. Note that 2.6 and 13/5 are examples of ways to enter a correct answer.

Question 40 40 of 569 selected Systems Of 2 Linear Equations In 2 Variables E

  • For the first line in the system:
    • The line is horizontal.
    • The line passes through the following points:
      • (negative 4 comma 3)
      • (0 comma 3)
      • (2 comma 3)
  • For the second line in the system:
    • The line slants sharply up from left to right.
    • The line passes through the following points:
      • (negative 1 comma negative 4.5)
      • (0 comma negative 2)
      • (2 comma 3)

The graph of a system of linear equations is shown. What is the solution (x,y) to the system?

  1. (0,3)

  2. (1,3)

  3. (2,3)

  4. (3,3)

Show Answer Correct Answer: C

Choice C is correct. The solution to this system of linear equations is represented by the point that lies on both lines shown, or the point of intersection of the two lines. According to the graph, the point of intersection occurs when x = 2 and y = 3 , or at the point (2,3). Therefore, the solution (x,y) to the system is (2,3).

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 41 41 of 569 selected Linear Equations In 1 Variable E

If 3 x - 8 = 7 , what is the value of 3 x + 8 ?

  1. -1

  2. 5

  3. 13

  4. 23

Show Answer Correct Answer: D

Choice D is correct. It's given that 3x-8=7. Adding 8 to both sides of this equation yields 3x=15. Adding 8 to both sides of this equation yields 3x+8=23. Therefore, the value of 3x+8 is 23.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect. This is the value of x, not 3x+8.

Choice C is incorrect and may result from conceptual or calculation errors.

Question 42 42 of 569 selected Linear Equations In 2 Variables H

Line p is defined by 4y+8x=6. Line r is perpendicular to line p in the xy-plane. What is the slope of line r ?

Show Answer Correct Answer: .5, 1/2

The correct answer is 12. For an equation in slope-intercept form y=mx+b, m represents the slope of the line in the xy-plane defined by this equation. It's given that line p is defined by 4y+8x=6. Subtracting 8x from both sides of this equation yields 4y=-8x+6. Dividing both sides of this equation by 4 yields y=-84x+64, or y=-2x+32. Thus, the slope of line p is -2 . If line r is perpendicular to line p , then the slope of line r is the negative reciprocal of the slope of line p . The negative reciprocal of -2 is -1(-2)=12. Note that 1/2 and .5 are examples of ways to enter a correct answer.

Question 43 43 of 569 selected Linear Equations In 1 Variable E

If 3 x - 27 = 24 , what is the value of x - 9 ?

  1. 1

  2. 8

  3. 24

  4. 35

Show Answer Correct Answer: B

Choice B is correct. Dividing each side of the given equation by 3 yields x - 9 = 8 . Therefore, the value of x - 9 is 8 .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect. This is the value of 3 x - 27 , not x - 9 .

Choice D is incorrect and may result from conceptual or calculation errors.

Question 44 44 of 569 selected Linear Equations In 2 Variables M

Line p is defined by 2y+18x=9. Line r is perpendicular to line p in the xy-plane. What is the slope of line r ?

  1. -9

  2. - 1 9

  3. 1 9

  4. 9

Show Answer Correct Answer: C

Choice C is correct. It’s given that line r is perpendicular to line p in the xy-plane. This means that the slope of line r is the negative reciprocal of the slope of line p . If the equation for line p is rewritten in slope-intercept form y = m x + b , where m and b are constants, then m is the slope of the line and (0,b) is its y-intercept. Subtracting 18 x from both sides of the equation 2y+18x=9 yields 2 y = - 18 x + 9 . Dividing both sides of this equation by 2 yields y = - 9 x + 9 2 . It follows that the slope of line p is -9 . The negative reciprocal of a number is -1 divided by the number. Therefore, the negative reciprocal of -9 is -1-9, or 1 9 . Thus, the slope of line r is 1 9 .

Choice A is incorrect. This is the slope of line p , not line r .

Choice B is incorrect. This is the reciprocal, not the negative reciprocal, of the slope of line p .

Choice D is incorrect. This is the negative, not the negative reciprocal, of the slope of line p .

Question 45 45 of 569 selected Linear Equations In 2 Variables M

  • The line slants gradually down from left to right.
  • The line passes through the following points:
    • (8 comma 10)
    • (9 comma 9)

The graph in the xy-plane models the possible combinations of length x , in meters (m), and width y , in meters, for a rectangle with a perimeter of 36 m. Which statement is the best interpretation of the point (8,10) in this context?

  1. The length is 10 m less than the perimeter, and the width is 8 m less than the perimeter.

  2. The length is 10 m, and the width is 8 m.

  3. The length is 8 m, and the width is 10 m.

  4. The length is 8 m less than the perimeter, and the width is 10 m less than the perimeter.

Show Answer Correct Answer: C

Choice C is correct. It’s given that the graph in the xy-plane models the possible combinations of length x, in meters (m), and width y, in meters, for a rectangle with a perimeter of 36 m. Since x represents the length, in meters, and y represents the width, in meters, the point (8,10) in the xy-plane represents a rectangle whose length is 8 m and whose width is 10 m.

Choice A is incorrect and may result from conceptual errors.

Choice B is incorrect. This is an interpretation of the point (10,8), not (8,10).

Choice D is incorrect and may result from conceptual errors.

Question 46 46 of 569 selected Linear Functions E

The function f is defined by f(x)=710x+55. What is the value of f(20)?

Show Answer Correct Answer: 69

The correct answer is 69. The value of f(20) can be found by evaluating the function f(x)=710x+55 for x=20. Substituting 20 for x in this function yields f(20)=710(20)+55, or f(20)=69. Therefore, the value of f(20) is 69.

Question 47 47 of 569 selected Linear Equations In 1 Variable E

8x-7x+130=260

What value of x is the solution to the given equation?

Show Answer Correct Answer: 130

The correct answer is 130 . It’s given that 8x-7x+130=260. Combining like terms on the left-hand side of this equation yields x+130=260. Subtracting 130 from each side of this equation yields x = 130 . Therefore, the value of x that's the solution to the given equation is 130 .

Question 48 48 of 569 selected Linear Inequalities In 1 Or 2 Variables H

Adam’s school is a 20-minute walk or a 5-minute bus ride away from his house. The bus runs once every 30 minutes, and the number of minutes, w, that Adam waits for the bus varies between 0 and 30. Which of the following inequalities gives the values of w for which it would be faster for Adam to walk to school?

  1. w minus 5, is less than 20
  2. w minus 5, is greater than 20
  3. w plus 5, is less than 20
  4. w plus 5, is greater than 20
Show Answer Correct Answer: D

Choice D is correct. It is given that w is the number of minutes that Adam waits for the bus. The total time it takes Adam to get to school on a day he takes the bus is the sum of the minutes, w, he waits for the bus and the 5 minutes the bus ride takes; thus, this time, in minutes, is w + 5. It is also given that the total amount of time it takes Adam to get to school on a day that he walks is 20 minutes. Therefore, w + 5 > 20 gives the values of w for which it would be faster for Adam to walk to school.

Choices A and B are incorrect because w – 5 is not the total length of time for Adam to wait for and then take the bus to school. Choice C is incorrect because the inequality should be true when walking 20 minutes is faster than the time it takes Adam to wait for and ride the bus, not less.

Question 49 49 of 569 selected Linear Functions E

If f is the function defined by f of x equals, the fraction with numerator 2 x minus 1, and denominator 3, what is the value of f of 5 ?

  1. 4 thirds

  2. 7 thirds

  3. 3

  4. 9

Show Answer Correct Answer: C

Choice C is correct. If f of x, equals the fraction with numerator 2 x minus 1, and denominator 3, then f of 5, equals the fraction with numerator 2 times 5, minus 1, and denominator 3, which equals the fraction with numerator 10 minus 1, and denominator 3, which equals the fraction 9 over 3, which equals 3.

Choice A is incorrect and may result from not multiplying x by 2 in the numerator. Choice B is incorrect and may result from dividing 2x by 3 and then subtracting 1. Choice D is incorrect and may result from evaluating only the numerator 2x – 1.

 

Question 50 50 of 569 selected Linear Equations In 1 Variable E

w + 7 = 357

What value of w is the solution to the given equation?

  1. 51

  2. 350

  3. 364

  4. 3,577

Show Answer Correct Answer: B

Choice B is correct. Subtracting 7 from each side of the given equation yields w = 350 . Therefore, the value of w that is the solution to the given equation is 350 .

Choice A is incorrect. This is the value of w that is the solution to the equation 7 w = 357 , not w + 7 = 357 .

Choice C is incorrect. This is the value of w that is the solution to the equation w - 7 = 357 , not w + 7 = 357 .

Choice D is incorrect and may result from conceptual or calculation errors.

Question 51 51 of 569 selected Linear Functions M

The function f is defined by f(x)=4x+k(x-1), where k is a constant, and f(5)=32. What is the value of f(10)?

Show Answer Correct Answer: 67

The correct answer is 67 . It’s given that f(5)=32. Therefore, for the given function f , when x = 5 , f(x)=32. Substituting 5 for x and 32 for f(x) in the given function f(x)=4x+k(x-1) yields 32=4(5)+k(5-1), or 32=20+4k. Subtracting 20 from each side of this equation yields 12=4k. Dividing each side of this equation by 4 yields k = 3 . Substituting 3 for k in the given function f(x)=4x+k(x-1) yields f(x)=4x+3(x-1), which is equivalent to f(x)=4x+3x-3, or f(x)=7x-3. Substituting 10 for x into this equation yields f(10)=7(10)-3, or f(10)=67. Therefore, the value of f(10) is 67 .

Question 52 52 of 569 selected Systems Of 2 Linear Equations In 2 Variables H

2 5 x + 7 5 y = 2 7

g x + k y = 5 2

In the given system of equations, g and k are constants. The system has infinitely many solutions. What is the value of g k ?

Show Answer Correct Answer: .2857, 2/7

The correct answer is 2 7 . It’s given that the system has infinitely many solutions. A system of two linear equations has infinitely many solutions if and only if the two linear equations are equivalent. Multiplying each side of the first equation in the system by 35 4 yields 354(25x+75y)=354(27), or 72x+494y=52. Since this equation is equivalent to the second equation and has the same right side as the second equation, the coefficients of x and y , respectively, should also be the same. It follows that g = 7 2 and k = 49 4 . Therefore, the value of g k is 72494, or 2 7 . Note that 2/7, .2857, 0.285, and 0.286 are examples of ways to enter a correct answer.

Question 53 53 of 569 selected Linear Inequalities In 1 Or 2 Variables M

2x-y>883

For which of the following tables are all the values of x and their corresponding values of y solutions to the given inequality?

Show Answer Correct Answer: D

Choice D is correct. All the tables in the choices have the same three values of x , 440 , 441 , and 442 , so each of the three values of x can be substituted in the given inequality to compare the corresponding values of y in each of the tables. Substituting 440 for x in the given inequality yields 2(440)-y>883, or 880-y>883. Subtracting 880 from both sides of this inequality yields -y>3. Dividing both sides of this inequality by -1 yields y<-3. Therefore, when x = 440 , the corresponding value of y must be less than -3 . Substituting 441 for x in the given inequality yields 2(441)-y>883, or 882-y>883. Subtracting 882 from both sides of this inequality yields -y>1. Dividing both sides of this inequality by -1 yields y<-1. Therefore, when x = 441 , the corresponding value of y must be less than -1 . Substituting 442 for x in the given inequality yields 2(442)-y>883, or 884-y>883. Subtracting 884 from both sides of this inequality yields -y>-1. Dividing both sides of this inequality by -1 yields y<1. Therefore, when x = 442 , the corresponding value of y must be less than 1 . For the table in choice D, when x = 440 , the corresponding value of y is -4 , which is less than -3 ; when x = 441 , the corresponding value of y is -2 , which is less than -1 ; when x = 442 , the corresponding value of y is 0 , which is less than 1 . Therefore, the table in choice D gives values of x and their corresponding values of y that are all solutions to the given inequality.

Choice A is incorrect. When x = 440 , the corresponding value of y in this table is 0 , which isn't less than -3 .

Choice B is incorrect. When x = 440 , the corresponding value of y in this table is 0 , which isn't less than -3 .

Choice C is incorrect. When x = 440 , the corresponding value of y in this table is -2 , which isn't less than -3 .

Question 54 54 of 569 selected Linear Functions E

d = 16 t

The given equation represents the distance d , in inches, where t represents the number of seconds since an object started moving. Which of the following is the best interpretation of 16 in this context?

  1. The object moved a total of 16 inches.

  2. The object moved a total of 16 t inches.

  3. The object is moving at a rate of 16 inches per second.

  4. The object is moving at a rate of 1 16 inches per second.

Show Answer Correct Answer: C

Choice C is correct. It’s given that in the equation d = 16 t , d represents the distance, in inches, and t represents the number of seconds since an object started moving. In this equation, t is being multiplied by 16 . This means that the object’s distance increases by 16 inches each second. Therefore, the best interpretation of 16 in this context is that the object is moving at a rate of 16 inches per second.

Choice A is incorrect and may result from conceptual errors.

Choice B is incorrect. This is the best interpretation of 16 t , rather than 16 , in this context.

Choice D is incorrect and may result from conceptual errors.

Question 55 55 of 569 selected Systems Of 2 Linear Equations In 2 Variables H

7 x + 6 y = 5

28 x + 24 y = 20

For each real number r , which of the following points lies on the graph of each equation in the xy-plane for the given system?

  1. (r,-6r7+57)

  2. (r,7r6+56)

  3. (r4+5,-r4+20)

  4. (-6r7+57,r)

Show Answer Correct Answer: D

Choice D is correct. Dividing each side of the second equation in the given system by 4 yields 7 x + 6 y = 5 . It follows that the two equations in the given system are equivalent and any point that lies on the graph of one equation will also lie on the graph of the other equation. Substituting r for y in the equation 7 x + 6 y = 5 yields 7x+6r=5. Subtracting 6 r from each side of this equation yields 7 x = - 6 r + 5 . Dividing each side of this equation by 7 yields x = - 6 r 7 + 5 7 . Therefore, the point (-6r7+57,r) lies on the graph of each equation in the xy-plane for each real number r .

Choice A is incorrect. Substituting r for x in the equation 7 x + 6 y = 5 yields 7 r + 6 y = 5 . Subtracting 7 r from each side of this equation yields 6 y = - 7 r + 5 . Dividing each side of this equation by 6 yields y = - 7 r 6 + 5 6 . Therefore, the point (r,-7r6+56), not the point (r,-6r7+57), lies on the graph of each equation.

Choice B is incorrect. Substituting r for x in the equation 7 x + 6 y = 5 yields 7 r + 6 y = 5 . Subtracting 7 r from each side of this equation yields 6 y = - 7 r + 5 . Dividing each side of this equation by 6 yields y = - 7 r 6 + 5 6 . Therefore, the point (r,-7r6+56), not the point (r,7r6+56), lies on the graph of each equation.

Choice C is incorrect. Substituting r 4 + 5 for x in the equation 7 x + 6 y = 5 yields 7(r4+5)+6y=5, or (7r4+35)+6y=5. Subtracting (7r4+35) from each side of this equation yields 6y=-7r4-35+5, or 6 y = - 7 r 4 - 30 . Dividing each side of this equation by 6 yields y = - 7 r 24 - 5 . Therefore, the point (r4+5,-7r24-5), not the point (r4+5,-r4+20), lies on the graph of each equation.

Question 56 56 of 569 selected Linear Equations In 1 Variable E

(p+3)+8=10

What value of p is the solution to the given equation?

  1. -1

  2. 5

  3. 15

  4. 21

Show Answer Correct Answer: A

Choice A is correct. Subtracting 8 from both sides of the given equation yields p + 3 = 2 . Subtracting 3 from both sides of this equation yields p = -1 .

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 57 57 of 569 selected Linear Functions M

f(x)=39

For the given linear function f , which table gives three values of x and their corresponding values of f(x)

Show Answer Correct Answer: B

Choice B is correct. For the given linear function f , f(x) must equal 39 for all values of x . Of the given choices, only choice B gives three values of x and their corresponding values of f(x) for the given linear function f .

Choice A is incorrect and may result from conceptual errors.

Choice C is incorrect and may result from conceptual errors.

Choice D is incorrect and may result from conceptual errors.

Question 58 58 of 569 selected Linear Equations In 1 Variable H

A manufacturing plant makes 10 -inch, 9 -inch, and 7 -inch frying pans. During a certain day, the number of 10 -inch frying pans that the manufacturing plant makes is 4 times the number n of 9 -inch frying pans it makes, and the number of 7 -inch frying pans it makes is 10 . During this day, the manufacturing plant makes 100 frying pans total. Which equation represents this situation?

  1. 10(4n)+9n+7(10)=100

  2. 10n+9n+7n=100

  3. 4n+10=100

  4. 5n+10=100

Show Answer Correct Answer: D

Choice D is correct. It's given that during a certain day, the number of 9-inch frying pans the manufacturing plant makes is n and the number of 7-inch frying pans it makes is 10. It's also given that during this day the number of 10-inch frying pans that the manufacturing plant makes is 4 times the number of 9-inch frying pans, or 4n. Therefore, the total number of 7-inch, 9-inch, and 10-inch frying pans the manufacturing plant makes is n+10+4n, or 5n+10. It's given that during this day the manufacturing plant makes 100 frying pans total. Thus, the equation 5n+10=100 represents this situation.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Question 59 59 of 569 selected Linear Equations In 2 Variables E

A mixture consisting of only vitamin D and calcium has a total mass of 150 grams. The mass of vitamin D in the mixture is 50 grams. What is the mass, in grams, of calcium in the mixture?

  1. 200

  2. 150

  3. 100

  4. 50

Show Answer Correct Answer: C

Choice C is correct. Let d represent the mass, in grams, of vitamin D in the mixture, and let c represent the mass, in grams, of calcium in the mixture. It’s given that the mixture consists of only vitamin D and calcium and that the total mass of the mixture is 150 grams. Therefore, the equation d+c=150 represents this situation. It’s also given that the mass of vitamin D in the mixture is 50 grams. Substituting 50 for d in the equation d+c=150 yields 50+c=150. Subtracting 50 from both sides of this equation yields c=100. Therefore, the mass of calcium in the mixture is 100 grams.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect. This is the total mass, in grams, of the mixture, not the mass, in grams, of calcium in the mixture.

Choice D is incorrect. This is the mass, in grams, of vitamin D in the mixture, not the mass, in grams, of calcium in the mixture.

Question 60 60 of 569 selected Linear Functions E

The function g is defined by g(x)=6x. For what value of x is g(x)=54 ?

Show Answer Correct Answer: 9

The correct answer is 9 . It’s given that g(x)=6x. Substituting 54 for g(x) in the given function yields 54 = 6 x . Dividing both sides of this equation by 6 yields x = 9 . Therefore, the value of x when g(x)=54 is 9 .

Question 61 61 of 569 selected Linear Equations In 1 Variable E

4 x + 5 = 165

What is the solution to the given equation?

Show Answer Correct Answer: 40

The correct answer is 40 . Subtracting 5 from both sides of the given equation yields 4x=160. Dividing both sides of this equation by 4 yields x=40. Therefore, the solution to the given equation is 40 .

Question 62 62 of 569 selected Linear Functions E

  • The line slants gradually up from left to right.
  • The line passes through the following points:
    • (negative 12 comma 0)
    • (negative 4 comma 2)

The graph of the linear function f is shown, where y=f(x). What is the x-intercept of the graph of f ?

  1. (-12,0)

  2. (0,0)

  3. (14,0)

  4. (12,0)

Show Answer Correct Answer: A

Choice A is correct. The x-intercept of a graph is the point where the graph intersects the x-axis. The graph of function f , where y=f(x), intersects the x-axis at (-12,0). Therefore, the x-intercept of the graph of f is (-12,0).

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 63 63 of 569 selected Systems Of 2 Linear Equations In 2 Variables M

A company that provides whale-watching tours takes groups of 21 people at a time. The company’s revenue is 80 dollars per adult and 60 dollars per child. If the company’s revenue for one group consisting of adults and children was 1,440 dollars, how many people in the group were children?

  1. 3

  2. 9

  3. 12

  4. 18

Show Answer Correct Answer: C

Choice C is correct. Let x represent the number of children in a whale-watching tour group. Let y represent the number of adults in this group. Because it's given that 21 people are in a group and the group consists of adults and children, it must be true that x+y=21. Since the company's revenue is 60 dollars per child, the total revenue from x children in this group was 60 x dollars. Since the company's revenue is 80 dollars per adult, the total revenue from y adults in this group was 80 y dollars. Because it's given that the total revenue for this group was 1,440 dollars, it must be true that 60x+80y=1,440. The equations x+y=21 and 60x+80y=1,440 form a linear system of equations that can be solved to find the value of x , which represents the number of children in the group, using the elimination method. Multiplying both sides of the equation x+y=21 by 80 yields 80x+80y=1,680. Subtracting 60x+80y=1,440 from 80x+80y=1,680 yields (80x+80y)-(60x+80y)=1,680-1,440, which is equivalent to 80x-60x+80y-80y=240, or 20x=240. Dividing both sides of this equation by 20 yields x=12. Therefore, 12 people in the group were children.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect. This is the number of adults in the group, not the number of children in the group.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 64 64 of 569 selected Linear Equations In 2 Variables M

Line k is defined by y=177x+4. Line j is parallel to line k in the xy-plane. What is the slope of line j ?

  1. 7 17

  2. 17 7

  3. 4

  4. 17

Show Answer Correct Answer: B

Choice B is correct. It's given that line k is defined by y = 17 x 7 + 4 . For an equation of a line written in the form y = m x + b , m is the slope of the line and b is the y-coordinate of the y-intercept of the line. It follows that the slope of line k is 17 7 . It's also given that line j is parallel to line k in the xy-plane. Since parallel lines have equal slopes, line j also has a slope of 17 7 .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect. This is the y-coordinate of the y-intercept of line k , not the slope of line j .

Choice D is incorrect and may result from conceptual or calculation errors.

Question 65 65 of 569 selected Systems Of 2 Linear Equations In 2 Variables H

- 12 x + 14 y = 36

- 6 x + 7 y = -18

How many solutions does the given system of equations have?

  1. Exactly one

  2. Exactly two

  3. Infinitely many

  4. Zero

Show Answer Correct Answer: D

Choice D is correct. A system of two linear equations in two variables, x and y , has zero solutions if the lines representing the equations in the xy-plane are distinct and parallel. Two lines are distinct and parallel if they have the same slope but different y-intercepts. Each equation in the given system can be written in slope-intercept form y=mx+b, where m is the slope of the line representing the equation in the xy-plane and (0,b) is the y-intercept. Adding 12 x to both sides of the first equation in the given system of equations, -12x+14y=36, yields 14y=12x+36. Dividing both sides of this equation by 14 yields y=67x+187. It follows that the first equation in the given system of equations has a slope of 67 and a y-intercept of (0,187). Adding 6 x to both sides of the second equation in the given system of equations, -6x+7y=-18, yields 7y=6x-18. Dividing both sides of this equation by 7 yields y=67x-187. It follows that the second equation in the given system of equations has a slope of 67 and a y-intercept of (0,-187). Since the slopes of these lines are the same and the y-intercepts are different, it follows that the given system of equations has zero solutions.

Alternate approach: To solve the system by elimination, multiplying the second equation in the given system of equations, -6x+7y=-18, by - 2 yields 12x-14y=36. Adding this equation to the first equation in the given system of equations, -12x+14y=36, yields (-12x+12x)+(-14y+14y)=36+36, or 0=72. Since this equation isn't true, the given system of equations has zero solutions.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Question 66 66 of 569 selected Systems Of 2 Linear Equations In 2 Variables H

In August, a car dealer completed 15 more than 3 times the number of sales the car dealer completed in September. In August and September, the car dealer completed 363 sales. How many sales did the car dealer complete in September?

Show Answer Correct Answer: 87

The correct answer is 87 . It’s given that in August, the car dealer completed 15 more than 3 times the number of sales the car dealer completed in September. Let x represent the number of sales the car dealer completed in September. It follows that 3 x + 15 represents the number of sales the car dealer completed in August. It’s also given that in August and September, the car dealer completed 363 sales. It follows that x+(3x+15)=363, or 4 x + 15 = 363 . Subtracting 15 from each side of this equation yields 4 x = 348 . Dividing each side of this equation by 4 yields x = 87 . Therefore, the car dealer completed 87 sales in September.

Question 67 67 of 569 selected Linear Functions M

In the xy-plane, the graph of the linear function f contains the points (0,3) and (7,31). Which equation defines f , where y=f(x)?

  1. f(x)=28x+34

  2. f(x)=3x+38

  3. f(x)=4x+3

  4. f(x)=7x+3

Show Answer Correct Answer: C

Choice C is correct. In the xy-plane, an equation of the graph of a linear function can be written in the form f(x)=mx+b, where m represents the slope and (0,b) represents the y-intercept of the graph of y=f(x). It’s given that the graph of the linear function f , where y=f(x), in the xy-plane contains the point (0,3). Thus, b = 3 . The slope of the graph of a line containing any two points (x1,y1) and (x2,y2) can be found using the slope formula, m=y2-y1x2-x1. Since it’s given that the graph of the linear function f contains the points (0,3) and (7,31), it follows that the slope of the graph of the line containing these points is m=31-37-0, or m = 4 . Substituting 4 for m and 3 for b in f(x)=mx+b yields f(x)=4x+3.

Choice A is incorrect. This function represents a graph with a slope of 28 and a y-intercept of (0,34).

Choice B is incorrect. This function represents a graph with a slope of 3 and a y-intercept of (0,38).

Choice D is incorrect. This function represents a graph with a slope of 7 and a y-intercept of (0,3).

Question 68 68 of 569 selected Systems Of 2 Linear Equations In 2 Variables E

y = 12 x - 20

y = 28

What is the solution (x,y) to the given system of equations?

  1. (4,28)

  2. (20,28)

  3. (28,4)

  4. (28,20)

Show Answer Correct Answer: A

Choice A is correct. The second equation in the given system is y = 28 . Substituting 28 for y in the first equation in the given system yields 28 = 12 x - 20 . Adding 20 to both sides of this equation yields 48 = 12 x . Dividing both sides of this equation by 12 yields 4 = x . Therefore, the solution (x,y) to the given system of equations is (4,28).

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect. This is the solution (y,x), not (x,y), to the given system of equations.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 69 69 of 569 selected Linear Functions E

  • The line slants gradually up from left to right.
  • The line passes through the following points:
    • (negative 5 comma 0)
    • (0 comma 2)

The graph of the linear function f is shown. What is the y-intercept of the graph of y=f(x)?

  1. (-5,0)

  2. (2,0)

  3. (0,2)

  4. (0,-5)

Show Answer Correct Answer: C

Choice C is correct. The y-intercept of a graph is the point where the graph intersects the y-axis. The graph of y=f(x) shown intersects the y-axis at the point (0,2). Therefore, the y-intercept of the graph of y=f(x) is (0,2).

Choice A is incorrect. This is the x-intercept, not the y-intercept, of the graph of y=f(x).

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 70 70 of 569 selected Linear Functions E

The number y is 84 less than the number x . Which equation represents the relationship between x and y ?

  1. y = x + 84

  2. y=184x

  3. y = 84 x

  4. y = x - 84

Show Answer Correct Answer: D

Choice D is correct. It’s given that the number y is 84 less than the number x . A number that's 84 less than the number x is equivalent to 84 subtracted from the number x , or x - 84 . Therefore, the equation y = x - 84 represents the relationship between x and y .

Choice A is incorrect and may result from conceptual errors.

Choice B is incorrect and may result from conceptual errors.

Choice C is incorrect and may result from conceptual errors.

Question 71 71 of 569 selected Linear Functions E

The function f is defined by f(x)=4x-3. What is the value of f(10)?

  1. -30

  2. 37

  3. 40

  4. 43

Show Answer Correct Answer: B

Choice B is correct. It’s given that the function f is defined by f(x)=4x-3. Substituting 10 for x in the given function yields f(10)=4(10)-3, which is equivalent to f(10)=40-3, or f(10)=37. Therefore, the value of f(10) is 37 .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect. This is the value of f(10) for the function f(x)=4x, not f(x)=4x-3.

Choice D is incorrect. This is the value of f(10) for the function f(x)=4x+3, not f(x)=4x-3.

Question 72 72 of 569 selected Linear Functions E

The function f is defined by the equation f(x)=7x+2. What is the value of f(x) when x = 4 ?

Show Answer Correct Answer: 30

The correct answer is 30 . The value of f(x) when x=4 can be found by substituting 4 for x in the given equation f(x)= 7x+2. This yields f(4)=7(4)+2, or f(4)=30. Therefore, when x=4, the value of f(x) is 30 .

Question 73 73 of 569 selected Systems Of 2 Linear Equations In 2 Variables M

At how many points do the graphs of the equations y = x + 20 and y = 8 x intersect in the xy-plane?

  1. 0

  2. 1

  3. 2

  4. 8

Show Answer Correct Answer: B

Choice B is correct. Each given equation is written in slope-intercept form, y = m x + b , where m is the slope and (0,b) is the y-intercept of the graph of the equation in the xy-plane. The graphs of two lines that have different slopes will intersect at exactly one point. The graph of the first equation is a line with slope 1 . The graph of the second equation is a line with slope 8 . Since the graphs are lines with different slopes, they will intersect at exactly one point.

Choice A is incorrect because two graphs of linear equations have 0 intersection points only if they are parallel and therefore have the same slope.

Choice C is incorrect because two graphs of linear equations in the xy-plane can have only 0 , 1 , or infinitely many points of intersection.

Choice D is incorrect because two graphs of linear equations in the xy-plane can have only 0 , 1 , or infinitely many points of intersection.

Question 74 74 of 569 selected Linear Inequalities In 1 Or 2 Variables M

Marisa needs to hire at least 10 staff members for an upcoming project. The staff members will be made up of junior directors, who will be paid $640 per week, and senior directors, who will be paid $880 per week. Her budget for paying the staff members is no more than $9,700 per week. She must hire at least 3 junior directors and at least 1 senior director. Which of the following systems of inequalities represents the conditions described if x is the number of junior directors and y is the number of senior directors?

  1. 640 x, plus 880 y, is greater than or equal to 9,700; x plus y is less than or equal to 10; x is greater than or equal to 3; y is greater than or equal to 1
  2. 640 x, plus 880 y, is less than or equal to 9,700; x plus y is greater than or equal to 10; x is greater than or equal to 3; y is greater than or equal to 1
  3. 640 x, plus 880 y, is greater than or equal to 9,700; x plus y is greater than or equal to 10; x is less than or equal to 3; y is less than or equal to 1
  4. 640 x, plus 880 y, is less than or equal to 9,700; x plus y is less than or equal to 10; x is less than or equal to 3; y is less than or equal to 1
Show Answer Correct Answer: B

Choice B is correct. Marisa will hire x junior directors and y senior directors. Since she needs to hire at least 10 staff members, x plus y, is greater than or equal to 10. Each junior director will be paid $640 per week, and each senior director will be paid $880 per week. Marisa’s budget for paying the new staff is no more than $9,700 per week; in terms of x and y, this condition is 640 x plus 880 y, is less than or equal to 9,700. Since Marisa must hire at least 3 junior directors and at least 1 senior director, it follows that x is greater than or equal to 3 and y is greater than or equal to 1. All four of these conditions are represented correctly in choice B.

Choices A and C are incorrect. For example, the first condition, 640 x plus 880 y, is greater than or equal to 9,700, in each of these options implies that Marisa can pay the new staff members more than her budget of $9,700. Choice D is incorrect because Marisa needs to hire at least 10 staff members, not at most 10 staff members, as the inequality x plus y, is less than or equal to 10 implies.

 

Question 75 75 of 569 selected Linear Inequalities In 1 Or 2 Variables M

For a snowstorm in a certain town, the minimum rate of snowfall recorded was 0.6 inches per hour, and the maximum rate of snowfall recorded was 1.8 inches per hour. Which inequality is true for all values of s , where s represents a rate of snowfall, in inches per hour, recorded for this snowstorm?

  1. s2.4

  2. s1.8

  3. 0s0.6

  4. 0.6s1.8

Show Answer Correct Answer: D

Choice D is correct. It's given that for a snowstorm in a certain town, the minimum rate of snowfall recorded was 0.6 inches per hour, the maximum rate of snowfall recorded was 1.8 inches per hour, and s represents a rate of snowfall, in inches per hour, recorded for this snowstorm. It follows that the inequality 0.6s1.8 is true for all values of s .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Question 76 76 of 569 selected Linear Inequalities In 1 Or 2 Variables E

During a portion of a flight, a small airplane's cruising speed varied between 150 miles per hour and 170 miles per hour. Which inequality best represents this situation, where s is the cruising speed, in miles per hour, during this portion of the flight? 

  1. s20

  2. s150

  3. s170

  4. 150s170

Show Answer Correct Answer: D

Choice D is correct. It's given that during a portion of a flight, a small airplane's cruising speed varied between 150 miles per hour and 170 miles per hour. It's also given that s represents the cruising speed, in miles per hour, during this portion of the flight. It follows that the airplane's cruising speed, in miles per hour, was at least 150 , which means s150, and was at most 170 , which means s170. Therefore, the inequality that best represents this situation is 150s170.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Question 77 77 of 569 selected Linear Equations In 1 Variable E

John paid a total of $165 for a microscope by making a down payment of $37 plus p monthly payments of $16 each. Which of the following equations represents this situation?

  1. 16 p - 37 = 165

  2. 37 p - 16 = 165

  3. 16 p + 37 = 165

  4. 37 p + 16 = 165

Show Answer Correct Answer: C

Choice C is correct. It’s given that John made a $16 payment each month for p months. The total amount of these payments can be represented by the expression 16 p . The down payment can be added to that amount to find the total amount John paid, yielding the expression 16 p + 37 . It’s given that John paid a total of $165. Therefore, the expression for the total amount John paid can be set equal to that amount, yielding the equation 16 p + 37 = 165 .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 78 78 of 569 selected Systems Of 2 Linear Equations In 2 Variables M

y = 3 x

2 x + y = 12

The solution to the given system of equations is (x,y). What is the value of 5 x ?

  1. 24

  2. 15

  3. 12

  4. 5

Show Answer Correct Answer: C

Choice C is correct. It's given by the first equation in the system that y=3x. Substituting 3 x for y in the equation 2x+y=12 yields 2x+3x=12, or 5x=12

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 79 79 of 569 selected Linear Inequalities In 1 Or 2 Variables E

Valentina bought two containers of beads. In the first container 30% of the beads are red, and in the second container 70% of the beads are red. Together, the containers have at least 400 red beads. Which inequality shows this relationship, where x is the total number of beads in the first container and y is the total number of beads in the second container?

  1. 0 point 3 x, plus, 0 point 7 y, is greater than or equal to, 400
  2. 0 point 7 x, plus, 0 point 3 y, is less than or equal to, 400
  3. the fraction x over 3, end fraction, plus, the fraction y over 7, end fraction, is less than or equal to, 400
  4. 30 x, plus, 70 y, is greater than or equal to, 400
Show Answer Correct Answer: A

Choice A is correct. It is given that x is the total number of beads in the first container and that 30% of those beads are red; therefore, the expression 0.3x represents the number of red beads in the first container. It is given that y is the total number of beads in the second container and that 70% of those beads are red; therefore, the expression 0.7y represents the number of red beads in the second container. It is also given that, together, the containers have at least 400 red beads, so the inequality that shows this relationship is 0.3x + 0.7y ≥ 400.

Choice B is incorrect because it represents the containers having a total of at most, rather than at least, 400 red beads. Choice C is incorrect and may be the result of misunderstanding how to represent a percentage of beads in each container. Also, the inequality shows the containers having a combined total of at most, rather than at least, 400 red beads. Choice D is incorrect because the percentages were not converted to decimals.

Question 80 80 of 569 selected Linear Equations In 2 Variables E

y = x + 4

Which table gives three values of x and their corresponding values of y for the given equation?

Show Answer Correct Answer: A

Choice A is correct. Substituting 0 for x into the given equation yields y=0+4, or y = 4 . Therefore, when x = 0 , the corresponding value of y for the given equation is 4 . Substituting 1 for x into the given equation yields y=1+4, or y = 5 . Therefore, when x = 1 , the corresponding value of y for the given equation is 5 . Substituting 2 for x into the given equation yields y=2+4, or y = 6 . Therefore, when x = 2 , the corresponding value of y for the given equation is 6 . Of the choices given, only the table in choice A gives these three values of x and their corresponding values of y for the given equation.

Choice B is incorrect. This table gives three values of x and their corresponding values of y for the equation y = - x + 6 .

Choice C is incorrect. This table gives three values of x and their corresponding values of y for the equation y = - x + 2 .

Choice D is incorrect. This table gives three values of x and their corresponding values of y for the equation y = x .

Question 81 81 of 569 selected Linear Functions E

As part of a science project on evaporation, Amaya measured the height of a liquid in a container over a period of time. The function f(x)=33-0.18x gives the estimated height, in centimeters (cm), of the liquid in the container x days after the start of the project. Which of the following is the best interpretation of 33 in this context?

  1. The estimated height, in cm, of the liquid at the start of the project

  2. The estimated height, in cm, of the liquid at the end of the project

  3. The estimated change in the height, in cm, of the liquid each day

  4. The estimated number of days for all of the liquid to evaporate

Show Answer Correct Answer: A

Choice A is correct. It's given that the function f(x)=33-0.18x gives the estimated height, in centimeters (cm), of the liquid in the container x days after the start of the project. For a linear function in the form f(x)=a+bx, where a and b are constants, a represents the value of f(0) and b represents the rate of change of the function. It follows that in the given function, 33 represents the value of f(0). Therefore, the best interpretation of 33 in this context is the estimated height, in cm, of the liquid at the start of the project.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect. The estimated change in the height, in cm, of the liquid each day is 0.18 , not 33 .

Choice D is incorrect and may result from conceptual or calculation errors.

Question 82 82 of 569 selected Linear Equations In 1 Variable E

Lorenzo purchased a box of cereal and some strawberries at the grocery store. Lorenzo paid $2 for the box of cereal and $1.90 per pound for the strawberries. If Lorenzo paid a total of $9.60 for the box of cereal and the strawberries, which of the following equations can be used to find p , the number of pounds of strawberries Lorenzo purchased? (Assume there is no sales tax.)

  1. 1.90p+2=9.60

  2. 1.90p-2=9.60

  3. 1.90+2p=9.60

  4. 1.90-2p=9.60

Show Answer Correct Answer: A

Choice A is correct. It's given that p represents the number of pounds of strawberries Lorenzo purchased and Lorenzo paid $1.90 per pound for the strawberries. It follows that the total amount, in dollars, Lorenzo paid for strawberries can be represented by 1.90p. It’s given that Lorenzo paid $2 for the box of cereal. If Lorenzo paid a total of $9.60 for the box of cereal and strawberries, it follows that the equation 1.90p+2=9.60 can be used to find p .

Choice B is incorrect and may result from conceptual errors.

Choice C is incorrect and may result from conceptual errors.

Choice D is incorrect and may result from conceptual errors.

Question 83 83 of 569 selected Linear Equations In 2 Variables M

24.5 x + 24.75 y = 641

Isabel ordered topsoil and crushed stone, which cost a total of $641 , for her garden. The given equation represents the relationship between the number of cubic yards of topsoil, x , and the number of tons of crushed stone, y , Isabel ordered. How much more, in dollars, did a ton of crushed stone cost Isabel than a cubic yard of topsoil?

Show Answer Correct Answer: 0.25, 1/4

The correct answer is .25. It’s given that the topsoil and crushed stone Isabel ordered for her garden cost a total of $641. It’s also given that the equation 24.5x+24.75y=641 represents the relationship between the number of cubic yards of topsoil, x, and the number of tons of crushed stone, y, that Isabel ordered. Since x represents the number of cubic yards of topsoil ordered, 24.5x represents the total cost, in dollars, of the topsoil, and the cost per cubic yard of topsoil is $24.50. Similarly, since y represents the number of tons of crushed stone ordered, 24.75y represents the total cost, in dollars, of crushed stone ordered, and the cost per ton of crushed stone is $24.75. Therefore, a ton of crushed stone cost Isabel 24.75-24.50, or 0.25, more dollars than a cubic yard of topsoil. Note that .25 and 1/4 are examples of ways to enter a correct answer.

Question 84 84 of 569 selected Systems Of 2 Linear Equations In 2 Variables H

48 x - 64 y = 48 y + 24

ry=18-12x

In the given system of equations, r is a constant. If the system has no solution, what is the value of r ?

Show Answer Correct Answer: -28

The correct answer is -28 . A system of two linear equations in two variables, x and y , has no solution if the lines represented by the equations in the xy-plane are distinct and parallel. The graphs of two lines in the xy-plane represented by equations in the form Ax+By=C, where A , B , and C are constants, are parallel if the coefficients for x and y in one equation are proportional to the corresponding coefficients for x and y in the other equation. The first equation in the given system, 48x-64y=48y+24, can be written in the form Ax+By=C by subtracting 48 y from both sides of the equation to yield 48x-112y=24. The second equation in the given system, ry=18-12x, can be written in the form Ax+By=C by adding 12 x to both sides of the equation to yield 12x+ry=18. The coefficient of x in the second equation is 14 times the coefficient of x in the first equation. That is, 48(14)=12. For the lines to be parallel, the coefficient of y in the second equation must also be 14 times the coefficient of y in the first equation. Therefore, -112(14)=r, or -28=r. Thus, if the given system has no solution, the value of r is -28 .

Question 85 85 of 569 selected Linear Equations In 2 Variables E

A chemist studying the impact of salt on a process mixes x kilograms of a low-salt mixture, which is 2 % salt by weight, with y kilograms of a high-salt mixture, which is 96 % salt by weight, to create 24 kilograms of a mixture that is 4 % salt by weight. Which equation represents this situation?

  1. 0.96x+0.02y=(0.04)(24)

  2. 0.02x+0.96y=(0.04)(24)

  3. 0.96x+0.02y=24

  4. 0.02x+0.96y=24

Show Answer Correct Answer: B

Choice B is correct. It’s given that a chemist mixes x kilograms of a low-salt mixture, which is 2% salt by weight. Multiplying 0.02 by the amount of the low-salt mixture, x kilograms, yields 0.02x kilograms of salt in the low-salt mixture. It's also given that the chemist mixes y kilograms of a high-salt mixture, which is 96% salt by weight. Multiplying 0.96 by the amount of the high-salt mixture, y kilograms, yields 0.96y kilograms of salt in the high-salt mixture. Therefore, the total amount of salt in the combined mixture is 0.02x+0.96y kilograms. It's given that the low-salt mixture and the high-salt mixture together create 24 kilograms of a combined mixture that is 4% salt by weight. Thus, the amount of salt in the combined mixture is 0.04(24) kilograms. Since the total amount of salt in the combined mixture equals the amount of salt in the low-salt mixture and the amount of salt in the high-salt mixture, the equation 0.02x+0.96y=(0.04)(24) represents this situation.

Choice A is incorrect. This equation represents a situation where the low-salt mixture is 96%, not 2%, salt by weight and the high-salt mixture is 2%, not 96%, salt by weight.

Choice C is incorrect. This equation represents a situation where the low-salt mixture is 96%, not 2%, salt by weight and the high-salt mixture is 2%, not 96%, salt by weight, and where the combined mixture contains 24 kilograms of salt, not 24 kilograms of a mixture that is 4% salt by weight.

Choice D is incorrect. This equation represents a situation where the combined mixture contains 24 kilograms of salt, not 24 kilograms of a mixture that is 4% salt by weight.

Question 86 86 of 569 selected Linear Inequalities In 1 Or 2 Variables E

The total cost, in dollars, to rent a surfboard consists of a $25 service fee and a $10 per hour rental fee. A person rents a surfboard for t hours and intends to spend a maximum of $75 to rent the surfboard. Which inequality represents this situation?

  1. 10t75

  2. 10+25t75

  3. 25t75

  4. 25+10t75

Show Answer Correct Answer: D

Choice D is correct. The cost of the rental fee depends on the number of hours the surfboard is rented. Multiplying t hours by 10 dollars per hour yields a rental fee of 10 t dollars. The total cost of the rental consists of the rental fee plus the 25 dollar service fee, which yields a total cost of 25+10t dollars. Since the person intends to spend a maximum of 75 dollars to rent the surfboard, the total cost must be at most 75 dollars. Therefore, the inequality 25+10t75 represents this situation.

Choice A is incorrect. This represents a situation where the rental fee, not the total cost, is at most 75 dollars.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Question 87 87 of 569 selected Linear Functions M

A model predicts that a certain animal weighed 241 pounds when it was born and that the animal gained 3 pounds per day in its first year of life. This model is defined by an equation in the form f(x)=a+bx, where f(x) is the predicted weight, in pounds, of the animal x days after it was born, and a and b are constants. What is the value of a ?

Show Answer Correct Answer: 241

The correct answer is 241 . For a certain animal, it's given that a model predicts the animal weighed 241 pounds when it was born and gained 3 pounds per day in its first year of life. It's also given that this model is defined by an equation in the form f(x)=a+bx, where f(x) is the predicted weight, in pounds, of the animal x days after it was born, and a and b are constants. It follows that a represents the predicted weight, in pounds, of the animal when it was born and b represents the predicted rate of weight gain, in pounds per day, in its first year of life. Thus, the value of a is 241 .

Question 88 88 of 569 selected Linear Equations In 2 Variables E

3 a plus 4 b, equals 25

A shipping company charged a customer $25 to ship some small boxes and some large boxes. The equation above represents the relationship between a, the number of small boxes, and b, the number of large boxes, the customer had shipped. If the customer had 3 small boxes shipped, how many large boxes were shipped?

  1. 3

  2. 4

  3. 5

  4. 6

Show Answer Correct Answer: B

Choice B is correct. It’s given that a represents the number of small boxes and b represents the number of large boxes the customer had shipped. If the customer had 3 small boxes shipped, then a, equals 3. Substituting 3 for a in the equation 3 a, plus 4 b, equals 25 yields 3 times 3, plus 4 b, equals 25 or 9 plus 4 b, equals 25. Subtracting 9 from both sides of the equation yields 4 b equals 16. Dividing both sides of this equation by 4 yields b equals 4. Therefore, the customer had 4 large boxes shipped.

Choices A, C, and D are incorrect. If the number of large boxes shipped is 3, then b equals 3. Substituting 3 for b in the given equation yields 3 a, plus, 4 times 3, equals 25 or 3 a, plus 12, equals 25. Subtracting 12 from both sides of the equation and then dividing by 3 yields a, equals thirteen thirds. However, it’s given that the number of small boxes shipped, a, is 3, not thirteen thirds, so b cannot equal 3. Similarly, if b equals 5 or b equals 6, then a, equals five thirds or a, equals one third, respectively, which is also not true.

 

Question 89 89 of 569 selected Linear Equations In 2 Variables E

What is the equation of the line that passes through the point (0,5) and is parallel to the graph of y = 7 x + 4 in the xy-plane?

  1. y = 5 x

  2. y = 7 x + 5

  3. y = 7 x

  4. y = 5 x + 7

Show Answer Correct Answer: B

Choice B is correct. The equation of a line in the xy-plane can be written in slope-intercept form y = m x + b , where m is the slope of the line and (0,b) is its y-intercept. It’s given that the line passes through the point (0,5). Therefore, b = 5 . It’s also given that the line is parallel to the graph of y = 7 x + 4 , which means the line has the same slope as the graph of y = 7 x + 4 . The slope of the graph of y = 7 x + 4 is 7 . Therefore, m = 7 . Substituting 7 for m and 5 for b in the equation y = m x + b yields y = 7 x + 5 .

Choice A is incorrect. The graph of this equation passes through the point (0,0), not (0,5), and has a slope of 5 , not 7 .

Choice C is incorrect. The graph of this equation passes through the point (0,0), not (0,5).

Choice D is incorrect. The graph of this equation passes through the point (0,7), not (0,5), and has a slope of 5 , not 7 .

Question 90 90 of 569 selected Systems Of 2 Linear Equations In 2 Variables M

A group of 202 people went on an overnight camping trip, taking 60 tents with them. Some of the tents held 2 people each, and the rest held 4 people each. Assuming all the tents were filled to capacity and every person got to sleep in a tent, exactly how many of the tents were 2-person tents?

  1. 30

  2. 20

  3. 19

  4. 18

Show Answer Correct Answer: C

Choice C is correct. Let x represent the number of 2-person tents and let y represent the number of 4-person tents. It is given that the total number of tents was 60 and the total number of people in the group was 202. This situation can be expressed as a system of two equations, x plus y, equals 60 and 2 x plus 4 y, equals 202. The first equation can be rewritten as y equals, negative x plus 60. Substituting negative x plus 60 for y in the equation 2 x plus 4 y, equals 202 yields 2 x plus, 4 times, open parenthesis, negative x plus 60, close parenthesis, equals 202. Distributing and combining like terms gives negative 2 x plus 240, equals 202. Subtracting 240 from both sides of negative 2 x plus 240, equals 202 and then dividing both sides by negative 2 gives x equals 19. Therefore, the number of 2-person tents is 19.

Alternate approach: If each of the 60 tents held 4 people, the total number of people that could be accommodated in tents would be 240. However, the actual number of people who slept in tents was 202. The difference of 38 accounts for the 2-person tents. Since each of these tents holds 2 people fewer than a 4-person tent, thirty eight halves, equals 19 gives the number of 2-person tents.

Choice A is incorrect. This choice may result from assuming exactly half of the tents hold 2 people. If that were true, then the total number of people who slept in tents would be 2 times 30, plus, 4 times 30, equals 180; however, the total number of people who slept in tents was 202, not 180. Choice B is incorrect. If 20 tents were 2-person tents, then the remaining 40 tents would be 4-person tents. Since all the tents were filled to capacity, the total number of people who slept in tents would be 2 times 20, plus, 4 times 40, equals, 40 plus 160, which equals 200; however, the total number of people who slept in tents was 202, not 200. Choice D is incorrect. If 18 tents were 2-person tents, then the remaining 42 tents would be 4-person tents. Since all the tents were filled to capacity, the total number of people who slept in tents would be 2 times 18, plus, 4 times 42, equals, 36 plus 168, which equals 204; however, the total number of people who slept in tents was 202, not 204.

Question 91 91 of 569 selected Linear Equations In 1 Variable M

A company that creates and sells tape dispensers calculates its monthly profit, in dollars, by subtracting its fixed monthly costs, in dollars, from its monthly sales revenue, in dollars. The equation 15,000=2.00x-4,500 represents this situation for a month where x tape dispensers are created and sold. Which statement is the best interpretation of 2.00x in this context?

  1. The monthly sales revenue, in dollars, from selling x tape dispensers

  2. The monthly sales revenue, in dollars, from each tape dispenser sold

  3. The monthly cost, in dollars, of creating each tape dispenser

  4. The monthly cost, in dollars, of creating x tape dispensers

Show Answer Correct Answer: A

Choice A is correct. It’s given that the equation 15,000=2.00x-4,500 represents this situation for a month where x tape dispensers are created and sold. It’s also given that the company calculates its monthly profit, in dollars, by subtracting its fixed monthly costs, in dollars, from its monthly sales revenue, in dollars. It follows that 2.00x represents the monthly sales revenue, in dollars. Therefore, the best interpretation of 2.00x in this context is the monthly sales revenue from selling x tape dispensers.

Choice B is incorrect. This is the best interpretation of 2.00, not 2.00x.

Choice C is incorrect and may result from conceptual errors.

Choice D is incorrect. This is the best interpretation of 4,500, not 2.00x.

Question 92 92 of 569 selected Systems Of 2 Linear Equations In 2 Variables E

A petting zoo sells two types of tickets. The standard ticket, for admission only, costs $5. The premium ticket, which includes admission and food to give to the animals, costs $12. One Saturday, the petting zoo sold a total of 250 tickets and collected a total of $2,300 from ticket sales. Which of the following systems of equations can be used to find the number of standard tickets, s, and premium tickets, p, sold on that Saturday?

  1. Equation 1: s plus p, equals 250. Equation 2: 5 s plus 12 p, equals 2,300.

  2. Equation 1: s plus p, equals 250. Equation 2: 12 s plus 5 p, equals 2,300.

  3. Equation 1: 5 s plus 12 p, equals 250. Equation 2: s plus p, equals 2,300.

  4. Equation 1: 12 s plus 5 p, equals 250. Equation 2: s plus p, equals 2,300.

Show Answer Correct Answer: A

Choice A is correct. It’s given that the petting zoo sells two types of tickets, standard and premium, and that s represents the number of standard tickets sold and p represents the number of premium tickets sold. It’s also given that the petting zoo sold 250 tickets on one Saturday; thus, s plus p, equals 250. It’s also given that each standard ticket costs $5 and each premium ticket costs $12. Thus, the amount collected in ticket sales can be represented by 5 s for standard tickets and 12 p for premium tickets. On that Saturday the petting zoo collected a total of $2,300 from ticket sales; thus, 5 s plus 12 p, equals 2,300. These two equations are correctly represented in choice A.

Choice B is incorrect. The second equation in the system represents the cost per standard ticket as $12, not $5, and the cost per premium ticket as $5, not $12. Choices C and D are incorrect. The equations represent the total collected from standard and premium ticket sales as $250, not $2,300, and the total number of standard and premium tickets sold as $2,300, not $250. Additionally, the first equation in choice D represents the cost per standard ticket as $12, not $5, and the cost per premium ticket as $5, not $12.

 

Question 93 93 of 569 selected Linear Inequalities In 1 Or 2 Variables M

In North America, the standard width of a parking space is at least 7.5 feet and no more than 9.0 feet. A restaurant owner recently resurfaced the restaurant’s parking lot and wants to determine the number of parking spaces, n, in the parking lot that could be placed perpendicular to a curb that is 135 feet long, based on the standard width of a parking space. Which of the following describes all the possible values of n ?

  1. 18 is less than or equal to n, which is less than or equal to 135

  2. 7 point 5 is less than or equal to n, which is less than or equal to 9

  3. 15 is less than or equal to n, which is less than or equal to 135

  4. 15 is less than or equal to n, which is less than or equal to 18

Show Answer Correct Answer: D

Choice D is correct. Placing the parking spaces with the minimum width of 7.5 feet gives the maximum possible number of parking spaces. Thus, the maximum number that can be placed perpendicular to a 135-foot-long curb is 135 over 7 point 5 equals 18. Placing the parking spaces with the maximum width of 9 feet gives the minimum number of parking spaces. Thus, the minimum number that can be placed perpendicular to a 135-foot-long curb is 135 over 9 equals 15. Therefore, if n is the number of parking spaces in the lot, the range of possible values for n is 15 is less than or equal to n, which is less than or equal to 18.

Choices A and C are incorrect. These choices equate the length of the curb with the maximum possible number of parking spaces. Choice B is incorrect. This is the range of possible values for the width of a parking space instead of the range of possible values for the number of parking spaces.

 

Question 94 94 of 569 selected Systems Of 2 Linear Equations In 2 Variables M

Two customers purchased the same kind of bread and eggs at a store. The first customer paid 12.45 dollars for 1 loaf of bread and 2 dozen eggs. The second customer paid 19.42 dollars for 4 loaves of bread and 1 dozen eggs. What is the cost, in dollars, of 1 dozen eggs?

  1. 3.77

  2. 3.88

  3. 4.15

  4. 4.34

Show Answer Correct Answer: D

Choice D is correct. Let l represent the cost, in dollars, of 1 loaf of bread, and let d represent the cost, in dollars, of 1 dozen eggs. It’s given that the first customer paid 12.45 dollars for 1 loaf of bread and 2 dozen eggs. Therefore, the first customer’s purchase can be represented by the equation l+2d=12.45. It’s also given that the second customer paid 19.42 dollars for 4 loaves of bread and 1 dozen eggs. Therefore, the second customer’s purchase can be represented by the equation 4l+d=19.42. The equations l+2d=12.45 and 4l+d=19.42 form a system of linear equations, which can be solved by elimination to find the value of d . Multiplying the first equation in the system by -4 yields -4l-8d=-49.8. Adding -4l-8d=-49.8 to the second equation, 4l+d=19.42, yields (-4l+4l)+(-8d+d)=(-49.8+19.42), which is equivalent to - 7 d = -30.38 . Dividing both sides of this equation by -7 yields d = 4.34 . Therefore, the cost, in dollars, of 1 dozen eggs is 4.34 .

Choice A is incorrect. This is the cost, in dollars, of 1 loaf of bread.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Question 95 95 of 569 selected Linear Equations In 2 Variables E

x plus y, equals 75

The equation above relates the number of minutes, x, Maria spends running each day and the number of minutes, y, she spends biking each day. In the equation, what does the number 75 represent?

  1. The number of minutes spent running each day

  2. The number of minutes spent biking each day

  3. The total number of minutes spent running and biking each day

  4. The number of minutes spent biking for each minute spent running

Show Answer Correct Answer: C

Choice C is correct. Maria spends x minutes running each day and y minutes biking each day. Therefore, x plus y represents the total number of minutes Maria spent running and biking each day. Because x plus y equals 75, it follows that 75 is the total number of minutes that Maria spent running and biking each day.

Choices A and B are incorrect. The number of minutes Maria spent running each day is represented by x and need not be 75. Similarly, the number of minutes that Maria spends biking each day is represented by y and need not be 75. The number of minutes Maria spends running each day and biking each day may vary; however, the total number of minutes she spends each day on these activities is constant and equal to 75. Choice D is incorrect. The number of minutes Maria spent biking for each minute spent running cannot be determined from the information provided.

 

Question 96 96 of 569 selected Linear Equations In 1 Variable E

3x+21=3x+k

In the given equation, k is a constant. The equation has infinitely many solutions. What is the value of k ?

Show Answer Correct Answer: 21

The correct answer is 21 . It's given that the equation 3 x + 21 = 3 x + k has infinitely many solutions. If an equation in one variable has infinitely many solutions, then the equation is true for any value of the variable. Subtracting 3 x from both sides of the given equation yields k = 21 . Since this equation must be true for any value of x , the value of k is 21 .

Question 97 97 of 569 selected Linear Functions E

g(x)=11x+4

For the given linear function g , which table shows three values of x and their corresponding values of g(x)?

Show Answer Correct Answer: C

Choice C is correct. Each of the tables shows the same three values of x : - 1 , 0 , and 1 . Substituting - 1 for x in the given function yields g(-1)=11(-1)+4, or g(-1)=-7. Therefore, when x=-1, the corresponding value of g(x) is - 7 . Substituting 0 for x in the given function yields g(0)=11(0)+4, or g(0)=4. Therefore, when x=0, the corresponding value of g(x) is 4 . Substituting 1 for x in the given function yields g(1)=11(1)+4, or g(1)=15. Therefore, when x=1, the corresponding value of g(x) is 15 . The table in choice C shows - 7 , 4 , and 15 as the corresponding value of g(x) for x-values of - 1 , 0 , and 1 , respectively. Therefore, the table in choice C shows three values of x and their corresponding values of g(x).

Choice A is incorrect. This table shows three values of x and their corresponding values of g(x) for the linear function g(x)=4x+11.

Choice B is incorrect. This table shows three values of x and their corresponding values of g(x) for the linear function g(x)=4x.

Choice D is incorrect. This table shows three values of x and their corresponding values of g(x) for the linear function g(x)=11x.

Question 98 98 of 569 selected Linear Equations In 1 Variable M

3 times, open parenthesis, 2 x minus 6, close parenthesis, minus 11, equals, 4 times, open parenthesis, x minus 3, close parenthesis, plus 6

If x is the solution to the equation above, what is the value of x minus 3 ?

  1. 23 over 2
  2. 17 over 2

  3. 15 over 2

  4. negative, 15 over 2

Show Answer Correct Answer: B

Choice B is correct. Because 2 is a factor of both 2 x and 6, the expression 2 x minus 6 can be rewritten as 2 times, open parenthesis, x minus 3, close parenthesis. Substituting 2 times, open parenthesis, x minus 3, close parenthesis for 2 x minus 6 on the left-hand side of the given equation yields 3 times 2, times, open parenthesis, x minus 3, close parenthesis, minus 11, equals, 4 times, open parenthesis, x minus 3, close parenthesis, plus 6, or 6 times, open parenthesis, x minus 3, close parenthesis, minus 11, equals, 4 times, open parenthesis, x minus 3, close parenthesis, plus 6. Subtracting 4 times, open parenthesis, x minus 3, close parenthesis from both sides of this equation yields 2 times, open parenthesis, x minus 3, close parenthesis, minus 11, equals 6. Adding 11 to both sides of this equation yields 2 times, open parenthesis, x minus 3, close parenthesis, equals 17. Dividing both sides of this equation by 2 yields x minus 3, equals, the fraction 17 over 2.

Alternate approach: Distributing 3 to the quantity 2 x minus 6 on the left-hand side of the given equation and distributing 4 to the quantity x minus 3 on the right-hand side yields 6 x minus 18, minus 11, equals, 4 x minus 12, plus 6, or 6 x minus 29, equals, 4 x minus 6. Subtracting 4 x from both sides of this equation yields 2 x minus 29, equals negative 6. Adding 29 to both sides of this equation yields 2 x equals 23. Dividing both sides of this equation by 2 yields x equals, the fraction 23 over 2. Therefore, the value of x minus 3 is the fraction 23 over 2, end fraction, minus 3, or the fraction 17 over 2.

Choice A is incorrect. This is the value of x, not x minus 3. Choices C and D are incorrect. If the value of x minus 3 is the fraction 15 over 2 or negative of the fraction 15 over 2, it follows that the value of x is the fraction 21 over 2 or negative of the fraction 9 over 2, respectively. However, solving the given equation for x yields x equals, the fraction 23 over 2. Therefore, the value of x minus 3 can’t be the fraction 15 over 2 or negative of the fraction 15 over 2.

 

Question 99 99 of 569 selected Systems Of 2 Linear Equations In 2 Variables E

y=4

x=y+6

The solution to the given system of equations is (x,y). What is the value of x ?

  1. 10

  2. 6

  3. 4

  4. 2

Show Answer Correct Answer: A

Choice A is correct. According to the first equation in the given system, y = 4 . Substituting 4 for y in the second equation in the given system yields x=4+6, or x = 10 .

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect. This is the value of y , not x .

Choice D is incorrect and may result from conceptual or calculation errors.

Question 100 100 of 569 selected Linear Equations In 1 Variable H

A factory makes 9 -inch, 7 -inch, and 4 -inch concrete screws. During a certain day, the number of 9 -inch concrete screws that the factory makes is 5 times the number n of 7 -inch concrete screws, and the number of 4 -inch concrete screws is 22 . During this day, the factory makes 100 concrete screws total. Which equation represents this situation?

  1. 9(5n)+7n+4(22)=100

  2. 9n+7n+4n=100

  3. 5n+22=100

  4. 6n+22=100

Show Answer Correct Answer: D

Choice D is correct. It's given that during a certain day at a factory, the number of 7 -inch concrete screws the factory makes is n and the number of 4 -inch concrete screws the factory makes is 22 . It's also given that during this day the number of 9 -inch concrete screws the factory makes is 5 times the number of 7 -inch concrete screws, or 5 n . Therefore, the total number of 7 -inch, 9 -inch, and 4 -inch concrete screws is n+5n+22, or 6 n + 22 . It's given that during this day, the factory makes 100 concrete screws total. Thus, the equation 6 n + 22 = 100 represents this situation.

Choice A is incorrect. This equation represents a situation where the total length, in inches, of all the concrete screws, rather than the total number of concrete screws, is 100 .

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect. This equation represents a situation where the total number of 9 -inch concrete screws and 4 -inch concrete screws, not including the 7 -inch concrete screws, is 100 .

Question 101 101 of 569 selected Systems Of 2 Linear Equations In 2 Variables E

  • For the first line in the system:
    • The line slants sharply down from left to right.
    • The line passes through the following points:
      • (0 comma 8)
      • (3 comma 4)
      • (5 comma four thirds)
  • For the second line in the system:
    • The line slants gradually up from left to right.
    • The line passes through the following points:
      • (0 comma 1)
      • (3 comma 4)
      • (5 comma 6)

The graph of a system of linear equations is shown. What is the solution (x,y) to the system?

  1. (2,3 )

  2. (3,4)

  3. (4,5 )

  4. (5,6 )

Show Answer Correct Answer: B

Choice B is correct. If a point (x,y) lies on both lines in the graph of a system of two linear equations, the ordered pair (x,y) is a solution to the system. The graph shown is the graph of a system of two linear equations, where the two lines in the graph intersect at the point (3,4). Therefore, the point (3,4) lies on both lines, so the ordered pair (3,4) is the solution to the system.

Choice A is incorrect. The point (2,3) lies on one, not both, of the lines in the graph shown.

Choice C is incorrect. The point (4,5) lies on one, not both, of the lines in the graph shown.

Choice D is incorrect. The point (5,6) lies on one, not both, of the lines in the graph shown.

Question 102 102 of 569 selected Linear Equations In 1 Variable H

Alan drives an average of 100 miles each week. His car can travel an average of 25 miles per gallon of gasoline. Alan would like to reduce his weekly expenditure on gasoline by $5. Assuming gasoline costs $4 per gallon, which equation can Alan use to determine how many fewer average miles, m, he should drive each week?

  1. the fraction 25 over 4, end fraction, times m, equals, 95

  2. the fraction 25 over 4, end fraction, times m, equals, 5

  3. the fraction 4 over 25, end fraction, times m, equals, 95

  4. the fraction 4 over 25, end fraction, times m, equals, 5

Show Answer Correct Answer: D

Choice D is correct. Since gasoline costs $4 per gallon, and since Alan’s car travels an average of 25 miles per gallon, the expression 4 over 25 gives the cost, in dollars per mile, to drive the car. Multiplying 4 over 25 by m gives the cost for Alan to drive m miles in his car. Alan wants to reduce his weekly spending by $5, so setting 4 over 25m equal to 5 gives the number of miles, m, by which he must reduce his driving.

Choices A, B, and C are incorrect. Choices A and B transpose the numerator and the denominator in the fraction. The fraction 25 over 4 would result in the unit miles per dollar, but the question requires a unit of dollars per mile. Choices A and C set the expression equal to 95 instead of 5, a mistake that may result from a misconception that Alan wants to reduce his driving by 5 miles each week; instead, the question says he wants to reduce his weekly expenditure by $5.

 

Question 103 103 of 569 selected Linear Equations In 2 Variables M

A certain township consists of a 5 -hectare industrial park and a 24 -hectare neighborhood. The total number of trees in the township is 4,529 . The equation 5 x + 24 y = 4,529 represents this situation. Which of the following is the best interpretation of x in this context?

  1. The average number of trees per hectare in the industrial park

  2. The average number of trees per hectare in the neighborhood

  3. The total number of trees in the industrial park

  4. The total number of trees in the neighborhood

Show Answer Correct Answer: A

Choice A is correct. It's given that a certain township consists of a 5 -hectare industrial park and a 24 -hectare neighborhood and that the total number of trees in the township is 4,529 . It's also given that the equation 5x+24y=4,529 represents this situation. Since the total number of trees for a given area can be determined by taking the size of the area, in hectares, times the average number of trees per hectare, the best interpretation of 5 x is the number of trees in the industrial park and the best interpretation of 24 y is the number of trees in the neighborhood. Since 5 is the size of the industrial park, in hectares, the best interpretation of x is the average number of trees per hectare in the industrial park.

Choice B is incorrect and may result from conceptual errors.

Choice C is incorrect and may result from conceptual errors.

Choice D is incorrect and may result from conceptual errors.

Question 104 104 of 569 selected Linear Inequalities In 1 Or 2 Variables M

  • The boundary of the inequality is a solid line.
    • The line slants sharply up from left to right.
    • The line passes through the following points:
      • (0 comma negative 4)
      • (1 comma 0)
      • (2 comma 4)
  • The area below and to the right of the boundary is shaded.

The shaded region shown represents the solutions to an inequality. Which ordered pair (x,y) is a solution to this inequality?

  1. (-5,-6)

  2. (-2,5)

  3. (1,4)

  4. (6,-2)

Show Answer Correct Answer: D

Choice D is correct. Since the shaded region shown represents the solutions to an inequality, an ordered pair (x,y) is a solution to the inequality if it's represented by a point in the shaded region. Of the given choices, only (6,-2) is represented by a point in the shaded region. Therefore, the ordered pair (6,-2) is a solution to this inequality.

Choice A is incorrect and may result from conceptual errors.

Choice B is incorrect and may result from conceptual errors.

Choice C is incorrect and may result from conceptual errors.

Question 105 105 of 569 selected Linear Equations In 2 Variables M

The equation 7 g + 7 b = 840 represents the number of blue tiles, b , and the number of green tiles, g , an artist needs for an 840 -square-inch tile project. The artist needs 71 blue tiles for the project. How many green tiles does he need?

Show Answer Correct Answer: 49

The correct answer is 49. It’s given that the equation 7g+7b=840 represents the number of blue tiles, b, and the number of green tiles, g, an artist needs for an 840-square-inch tile project. It’s also given that the artist needs 71 blue tiles for the project. Substituting 71 for b in the equation 7g+7b=840 yields 7g+7(71)=840, or 7g+497=840. Subtracting 497 from both sides of this equation yields 7g=343. Dividing both sides of this equation by 7 yields g=49. Therefore, the artist needs 49 green tiles for the project.

Question 106 106 of 569 selected Linear Inequalities In 1 Or 2 Variables H

A laundry service is buying detergent and fabric softener from its supplier. The supplier will deliver no more than 300 pounds in a shipment. Each container of detergent weighs 7.35  pounds, and each container of fabric softener weighs 6.2 pounds. The service wants to buy at least twice as many containers of detergent as containers of fabric softener. Let d represent the number of containers of detergent, and let s represent the number of containers of fabric softener, where d and s are nonnegative integers. Which of the following systems of inequalities best represents this situation?

  1. 7 point three five d, plus, 6 point 2 s, is less than or equal to 300, and, d is greater than or equal to 2 s
  2. 7 point three five d, plus, 6 point 2 s, is less than or equal to 300, and, 2 d is greater than or equal to s
  3. 14 point 7 d, plus, 6 point 2 s, is less than or equal to 300, and, d is greater than or equal to 2 s
  4. 14 point 7 d, plus, 6 point 2 s, is less than or equal to 300, and, 2 d is greater than or equal to s
Show Answer Correct Answer: A

Choice A is correct. The number of containers in a shipment must have a weight less than or equal to 300 pounds. The total weight, in pounds, of detergent and fabric softener that the supplier delivers can be expressed as the weight of each container multiplied by the number of each type of container, which is 7.35d for detergent and 6.2s for fabric softener. Since this total cannot exceed 300 pounds, it follows that 7 point 3 5 d, plus 6 point 2 s, is less than or equal to 300. Also, since the laundry service wants to buy at least twice as many containers of detergent as containers of fabric softener, the number of containers of detergent should be greater than or equal to two times the number of containers of fabric softener. This can be expressed by the inequality d is greater than or equal to 2 s.

Choice B is incorrect because it misrepresents the relationship between the numbers of each container that the laundry service wants to buy. Choice C is incorrect because the first inequality of the system incorrectly doubles the weight per container of detergent. The weight of each container of detergent is 7.35, not 14.7 pounds. Choice D is incorrect because it doubles the weight per container of detergent and transposes the relationship between the numbers of containers.

Question 107 107 of 569 selected Linear Functions E

The function g is defined by g of x, equals, negative x, plus 8. What is the value of g of 0?

  1. negative 8

  2. 0

  3. 4

  4. 8

Show Answer Correct Answer: D

Choice D is correct. The value of g of 0 is found by substituting 0 for x in the function g. This yields g of 0 equals, negative 0, plus 8, which can be rewritten as g of 0 equals 8.

Choice A is incorrect and may result from misinterpreting the equation as g of x equals, x plus, negative 8 instead of g of x equals, negative x plus, negative 8. Choice B is incorrect. This is the value of x, not g of x. Choice C is incorrect and may result from calculation errors.

 

Question 108 108 of 569 selected Linear Equations In 2 Variables M

A chemist combines water and acetic acid to make a mixture with a volume of 56 milliliters (mL). The volume of acetic acid in the mixture is 10 mL. What is the volume of water, in mL, in the mixture? (Assume that the volume of the mixture is the sum of the volumes of water and acetic acid before they were mixed.)

Show Answer Correct Answer: 46

The correct answer is 46 . It's given that a chemist combines water and acetic acid to make a mixture with a volume of 56 milliliters (mL) and that the volume of acetic acid in the mixture is 10 mL. Let x represent the volume of water, in mL, in the mixture. The equation x + 10 = 56 represents this situation. Subtracting 10 from both sides of this equation yields x = 46 . Therefore, the volume of water, in mL, in the mixture is 46 .

Question 109 109 of 569 selected Linear Inequalities In 1 Or 2 Variables M

Inequality 1: y is less than or equal to x.

Inequality 2: y is less than or equal to negative x.

Which of the following ordered pairs x comma y is a solution to the system of inequalities above?

  1. 1 comma 0

  2. negative 1 comma 0

  3. 0 comma 1

  4. 0 comma negative 1

Show Answer Correct Answer: D

Choice D is correct. The solutions to the given system of inequalities is the set of all ordered pairs x comma y that satisfy both inequalities in the system. For an ordered pair to satisfy the inequality y is less than or equal to x, the value of the ordered pair’s y-coordinate must be less than or equal to the value of the ordered pair’s x-coordinate. This is true of the ordered pair 0 comma negative 1, because negative 1 is less than or equal to 0. To satisfy the inequality y is less than or equal to negative x, the value of the ordered pair’s y-coordinate must be less than or equal to the value of the additive inverse of the ordered pair’s x-coordinate. This is also true of the ordered pair 0 comma negative 1. Because 0 is its own additive inverse, negative 1 is less than or equal to the negative of 0 is the same as negative 1 is less than or equal to 0. Therefore, the ordered pair 0 comma negative 1 is a solution to the given system of inequalities.

Choice A is incorrect. This ordered pair satisfies only the inequality y is less than or equal to x in the given system, not both inequalities. Choice B incorrect. This ordered pair satisfies only the inequality y is less than or equal to negative x in the system, but not both inequalities. Choice C is incorrect. This ordered pair satisfies neither inequality.

 

Question 110 110 of 569 selected Linear Equations In 2 Variables M

Line t in the xy-plane has a slope of - 1 3 and passes through the point (9,10). Which equation defines line t ?

  1. y = 13 x - 1 3

  2. y = 9 x + 10

  3. y = - x 3 + 10

  4. y = - x 3 + 13

Show Answer Correct Answer: D

Choice D is correct. The equation that defines line t in the xy-plane can be written in slope-intercept form y = m x + b , where m is the slope of line t and (0,b) is its y-intercept. It’s given that line t has a slope of - 1 3 . Therefore, m = - 1 3 . Substituting - 1 3 for m in the equation y = m x + b yields y=-13x+b, or y = - x 3 + b . It’s also given that line t passes through the point (9,10). Substituting 9 for x and 10 for y in the equation y = - x 3 + b yields 10=-93+b, or 10=-3+b. Adding 3 to both sides of this equation yields 13 = b . Substituting 13 for b in the equation y = - x 3 + b yields y = - x 3 + 13 .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect. This equation defines a line that has a slope of 9 , not - 1 3 , and passes through the point (0,10), not (9,10).

Choice C is incorrect. This equation defines a line that passes through the point (0,10), not (9,10).

Question 111 111 of 569 selected Linear Equations In 1 Variable M

2 n plus 6, equals 14

A tree had a height of 6 feet when it was planted. The equation above can be used to find how many years n it took the tree to reach a height of 14 feet. Which of the following is the best interpretation of the number 2 in this context?

  1. The number of years it took the tree to double its height

  2. The average number of feet that the tree grew per year

  3. The height, in feet, of the tree when the tree was 1 year old

  4. The average number of years it takes similar trees to grow 14 feet

Show Answer Correct Answer: B

Choice B is correct. The height of the tree at a given time is equal to its height when it was planted plus the number of feet that the tree grew. In the given equation, 14 represents the height of the tree at the given time, and 6 represents the height of the tree when it was planted. It follows that 2 n represents the number of feet the tree grew from the time it was planted until the time it reached a height of 14 feet. Since n represents the number of years between the given time and the time the tree was planted, 2 must represent the average number of feet the tree grew each year.

Choice A is incorrect and may result from interpreting the coefficient 2 as doubling instead of as increasing by 2 each year. Choice C is incorrect. The height of the tree when it was 1 year old was 2 times 1, plus 6, equals 8 feet, not 2 feet. Choice D is incorrect. No information is given to connect the growth of one particular tree to the growth of similar trees.

 

Question 112 112 of 569 selected Linear Equations In 1 Variable M

2x+16=a(x+8)

In the given equation, a is a constant. If the equation has infinitely many solutions, what is the value of a ?

Show Answer Correct Answer: 2

The correct answer is 2 . An equation with one variable, x , has infinitely many solutions only when both sides of the equation are equal for any defined value of x . It's given that 2x+16=a(x+8), where a is a constant. This equation can be rewritten as 2(x+8)=a(x+8). If this equation has infinitely many solutions, then both sides of this equation are equal for any defined value of x . Both sides of this equation are equal for any defined value of x when 2=a. Therefore, if the equation has infinitely many solutions, the value of a is 2 .

Alternate approach: If the given equation, 2x+16=a(x+8), has infinitely many solutions, then both sides of this equation are equal for any value of x . If x = 0 , then substituting 0 for x in 2x+16=a(x+8) yields 2(0)+16=a(0+8), or 16=8a. Dividing both sides of this equation by 8 yields 2=a.

Question 113 113 of 569 selected Linear Equations In 2 Variables M

In the xy-plane, line t passes through the points (0,9) and (1,17). Which equation defines line t ?

  1. y=18x+9

  2. y = x + 1 8

  3. y = x + 8

  4. y = 8 x + 9

Show Answer Correct Answer: D

Choice D is correct. An equation defining a line in the xy-plane can be written in the form y=mx+b, where m represents the slope and (0,b) represents the y-intercept of the line. It’s given that line t passes through the point (0,9); therefore, b = 9 . The slope, m , of a line can be found using any two points on the line, (x1,y1) and (x2,y2), and the slope formula m=y2-y1x2-x1. Substituting (0,9) and (1,17) for (x1,y1) and (x2,y2), respectively, in the slope formula yields m=17-91-0, or m=8. Substituting 8 for m and 9 for b in the equation y=mx+b yields y=8x+9.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Question 114 114 of 569 selected Linear Equations In 1 Variable E

If x 8 = 5 , what is the value of 8 x ?

Show Answer Correct Answer: .2, 1/5

The correct answer is 15. Since the number 5 can also be written as 51, the given equation can also be written as x8=51. This equation is equivalent to 8x=15. Therefore, the value of 8x is 15. Note that 1/5 and .2 are examples of ways to enter a correct answer.

Alternate approach: Multiplying both sides of the equation x8=5 by 8 yields x=40. Substituting 40 for x into the expression 8x yields 840, or 15

Question 115 115 of 569 selected Linear Equations In 2 Variables E

A machine makes large boxes or small boxes, one at a time, for a total of 700 minutes each day. It takes the machine 10 minutes to make a large box or 5 minutes to make a small box. Which equation represents the possible number of large boxes, x , and small boxes, y , the machine can make each day?

  1. 5x+10y=700

  2. 10x+5y=700

  3. (x+y)(10+5)=700

  4. (10+x)(5+y)=700

Show Answer Correct Answer: B

Choice B is correct. It’s given that it takes the machine 10 minutes to make a large box. It's also given that x represents the possible number of large boxes the machine can make each day. Multiplying 10 by x gives 10 x , which represents the amount of time spent making large boxes. It’s given that it takes the machine 5 minutes to make a small box. It's also given that y represents the possible number of small boxes the machine can make each day. Multiplying 5 by y gives 5 y , which represents the amount of time spent making small boxes. Combining the amount of time spent making x large boxes and y small boxes yields 10 x + 5 y . It’s given that the machine makes boxes for a total of 700 minutes each day. Therefore 10 x + 5 y = 700 represents the possible number of large boxes, x , and small boxes, y , the machine can make each day.

Choice A is incorrect and may result from associating the time of 10 minutes with small, rather than large, boxes and the time of 5 minutes with large, rather than small, boxes.

Choice C is incorrect and may result from conceptual errors.

Choice D is incorrect and may result from conceptual errors.

Question 116 116 of 569 selected Linear Equations In 1 Variable E

The perimeter of an isosceles triangle is 83 inches. Each of the two congruent sides of the triangle has a length of 24 inches. What is the length, in inches, of the third side?

Show Answer Correct Answer: 35

The correct answer is 35 . It’s given that the perimeter of an isosceles triangle is 83 inches and that each of the two congruent sides has a length of 24 inches. The perimeter of a triangle is the sum of the lengths of its three sides. The equation 24+24+x=83 can be used to represent this situation, where x is the length, in inches, of the third side. Combining like terms on the left-hand side of this equation yields 48+x=83. Subtracting 48 from both sides of this equation yields x = 35 . Therefore, the length, in inches, of the third side is 35.

Question 117 117 of 569 selected Linear Equations In 1 Variable E

If 3 x = 30 , what is the value of 3 x - 12 ?

  1. -2

  2. 18

  3. 22

  4. 42

Show Answer Correct Answer: B

Choice B is correct. Subtracting 12 from each side of the given equation yields 3x-12=30-12, or 3x-12=18. Therefore, the value of 3x-12 is 18 .

Choice A is incorrect. This is the value of x-12, not 3x-12.

Choice C is incorrect. This is the value of x+12, not 3x-12.

Choice D is incorrect. This is the value of 3x+12, not 3x-12.

Question 118 118 of 569 selected Linear Equations In 1 Variable M

open parenthesis, b minus 2, close parenthesis, times x, equals 8

In the given equation, b is a constant. If the equation has no solution, what is the value of b ?

  1. 2

  2. 4

  3. 6

  4. 10

Show Answer Correct Answer: A

Choice A is correct. This equation has no solution when there is no value of x that produces a true statement. Solving the given equation for x by dividing both sides by open parenthesis, b minus 2, close parenthesis gives x equals, the fraction with numerator 8, and denominator, open parenthesis, b minus 2, close parenthesis. When open parenthesis, b minus 2, close parenthesis, equals 0, the right-hand side of this equation will be undefined, and the equation will have no solution. Therefore, when b equals 2, there is no value of x that satisfies the given equation.

Choices B, C, and D are incorrect. Substituting 4, 6, and 10 for b in the given equation yields exactly one solution, rather than no solution, for x. For example, substituting 4 for b in the given equation yields open parenthesis, 4 minus 2, close parenthesis, times x, equals 8, or 2 x equals 8. Dividing both sides of 2 x equals 8 by 2 yields x equals 4. Similarly, if b equals 6 or b equals 10, x equals 2 and x equals 1, respectively.

 

Question 119 119 of 569 selected Linear Equations In 2 Variables E

Line r in the xy-plane has a slope of 4 and passes through the point (0,6). Which equation defines line r ?

  1. y = - 6 x + 4

  2. y = 6 x + 4

  3. y = 4 x - 6

  4. y = 4 x + 6

Show Answer Correct Answer: D

Choice D is correct. A line in the xy-plane with a slope of m and a y-intercept of (0,b) can be defined by an equation in the form y = m x + b . It’s given that line r has a slope of 4 and passes through the point (0,6). It follows that m = 4 and b = 6 . Substituting 4 for m and 6 for b in the equation y = m x + b yields y = 4 x + 6 . Therefore, the equation y = 4 x + 6 defines line r .

Choice A is incorrect. This equation defines a line that has a slope of -6 , not 4 , and passes through the point (0,4), not (0,6).

Choice B is incorrect. This equation defines a line that has a slope of 6 , not 4 , and passes through the point (0,4), not (0,6).

Choice C is incorrect. This equation defines a line that passes through the point (0,-6), not (0,6).

Question 120 120 of 569 selected Linear Equations In 2 Variables E

A total of 364 paper straws of equal length were used to construct two types of polygons: triangles and rectangles. The triangles and rectangles were constructed so that no two polygons had a common side. The equation 3 x + 4 y = 364 represents this situation, where x is the number of triangles constructed and y is the number of rectangles constructed. What is the best interpretation of (x,y)=(24,73) in this context?

  1. If 24 triangles were constructed, then 73 rectangles were constructed.

  2. If 24 triangles were constructed, then 73 paper straws were used.

  3. If 73 triangles were constructed, then 24 rectangles were constructed.

  4. If 73 triangles were constructed, then 24 paper straws were used.

Show Answer Correct Answer: A

Choice A is correct. It's given that 364 paper straws of equal length were used to construct triangles and rectangles, where no two polygons had a common side. It's also given that the equation 3x+4y=364 represents this situation, where x is the number of triangles constructed and y is the number of rectangles constructed. The equation (x,y)=(24,73) means that if x=24, then y=73. Substituting 24 for x and 73 for y in 3x+4y=364 yields 3(24)+4(73)=364, or 364=364, which is true. Therefore, in this context, the equation (x,y)=(24,73) means that if 24 triangles were constructed, then 73 rectangles were constructed. 

Choice B is incorrect and may result from conceptual errors.

Choice C is incorrect and may result from conceptual errors.

Choice D is incorrect and may result from conceptual errors.

Question 121 121 of 569 selected Linear Equations In 2 Variables M

If the graph of 27 x + 33 y = 297 is shifted down 5 units in the xy-plane, what is the y-intercept of the resulting graph?

  1. (0,4)

  2. (0,6)

  3. (0,14)

  4. (0,28)

Show Answer Correct Answer: A

Choice A is correct. When the graph of an equation in the form A x + B y = C , where A , B , and C are constants, is shifted down k units in the xy-plane, the resulting graph can be represented by the equation Ax+B(y+k)=C. It's given that the graph of 27 x + 33 y = 297 is shifted down 5 units in the xy-plane. Therefore, the resulting graph can be represented by the equation 27x+33(y+5)=297, or 27 x + 33 y + 165 = 297 . Subtracting 165 from both sides of this equation yields 27 x + 33 y = 132 . The y-intercept of the graph of an equation in the xy-plane is the point where the line intersects the y-axis, represented by the point (0,y). Substituting 0 for x in the equation 27 x + 33 y = 132 yields 27(0)+33y=132, or 33 y = 132 . Dividing both sides of this equation by 33 yields y = 4 . Therefore, if the graph of 27 x + 33 y = 297 is shifted down 5 units, the y-intercept of the resulting graph is (0,4).

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect. This is the y-intercept of the graph of 27 x + 33 y = 297 shifted up, not down, 5 units.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 122 122 of 569 selected Linear Functions E

The function f is defined by f(x)=4x. For what value of x does f(x)=8?

Show Answer Correct Answer: 2

The correct answer is 2 . Substituting 8 for f(x) in the given equation yields 8=4x. Dividing the left- and right-hand sides of this equation by 4 yields x=2. Therefore, the value of x is 2 when f(x)=8.

Question 123 123 of 569 selected Linear Inequalities In 1 Or 2 Variables E

On a car trip, Rhett and Jessica each drove for part of the trip, and the total distance they drove was under 220 miles. Rhett drove at an average speed of 35 miles per hour (mph), and Jessica drove at an average speed of 40 mph. Which of the following inequalities represents this situation, where r is the number of hours Rhett drove and j is the number of hours Jessica drove?

  1. 35r+40j>220

  2. 35r+40j<220

  3. 40r+35j>220

  4. 40r+35j<220

Show Answer Correct Answer: B

Choice B is correct. It’s given that Rhett drove at an average speed of 35 miles per hour and that he drove for r hours. Multiplying 35 miles per hour by r hours yields 35 r miles, or the distance that Rhett drove. It’s also given that Jessica drove at an average speed of 40 miles per hour and that she drove for j hours. Multiplying 40 miles per hour by j hours yields 40j miles, or the distance that Jessica drove. The total distance, in miles, that Rhett and Jessica drove can be represented by the expression 35r+40j. It’s given that the total distance they drove was under 220 miles. Therefore, the inequality 35r+40j<220 represents this situation.

Choice A is incorrect. This inequality represents a situation in which the total distance Rhett and Jessica drove was over, rather than under, 220 miles.

Choice C is incorrect. This inequality represents a situation in which Rhett drove at an average speed of 40 , rather than 35 , miles per hour, Jessica drove at an average speed of 35 , rather than 40 , miles per hour, and the total distance they drove was over, rather than under, 220 miles.

Choice D is incorrect. This inequality represents a situation in which Rhett drove at an average speed of 40 , rather than 35 , miles per hour, and Jessica drove at an average speed of 35 , rather than 40 , miles per hour.

Question 124 124 of 569 selected Linear Inequalities In 1 Or 2 Variables H

A local transit company sells a monthly pass for $95 that allows an unlimited number of trips of any length. Tickets for individual trips cost $1.50, $2.50, or $3.50, depending on the length of the trip. What is the minimum number of trips per month for which a monthly pass could cost less than purchasing individual tickets for trips?

Show Answer

The correct answer is 28. The minimum number of individual trips for which the cost of the monthly pass is less than the cost of individual tickets can be found by assuming the maximum cost of the individual tickets, $3.50. If n tickets costing $3.50 each are purchased in one month, the inequality 95 < 3.50n represents this situation. Dividing both sides of the inequality by 3.50 yields 27.14 < n, which is equivalent to n > 27.14. Since only a whole number of tickets can be purchased, it follows that 28 is the minimum number of trips. 

Question 125 125 of 569 selected Linear Equations In 2 Variables E

The equation y equals, 0 point 1 x models the relationship between the number of different pieces of music a certain pianist practices, y, during an x-minute practice session. How many pieces did the pianist practice if the session lasted 30 minutes?

  1. 1

  2. 3

  3. 10

  4. 30

Show Answer Correct Answer: B

Choice B is correct. It’s given that the equation y equals, 0 point 1 x models the relationship between the number of different pieces of music a certain pianist practices, y, and the number of minutes in a practice session, x. Since it’s given that the session lasted 30 minutes, the number of pieces the pianist practiced can be found by substituting 30 for x in the given equation, which yields y equals, 0 point 1 times 30, or y equals 3.

Choices A and C are incorrect and may result from misinterpreting the values in the equation. Choice D is incorrect. This is the given value of x, not the value of y.

Question 126 126 of 569 selected Linear Inequalities In 1 Or 2 Variables E

A bakery sells trays of cookies. Each tray contains at least 50 cookies but no more than 60. Which of the following could be the total number of cookies on 4 trays of cookies?

  1. 165

  2. 205

  3. 245

  4. 285

Show Answer Correct Answer: B

Choice B is correct. If each tray contains the least number of cookies possible, 50 cookies, then the least number of cookies possible on 4 trays is 50 × 4 = 200 cookies. If each tray contains the greatest number of cookies possible, 60 cookies, then the greatest number of cookies possible on 4 trays is 60 × 4 = 240 cookies. If the least number of cookies on 4 trays is 200 and the greatest number of cookies is 240, then 205 could be the total number of cookies on these 4 trays of cookies because 200 is less than or equal to 205, which is less than or equal to 240..

Choices A, C, and D are incorrect. The least number of cookies on 4 trays is 200 cookies, and the greatest number of cookies on 4 trays is 240 cookies. The choices 165, 245, and 285 are each either less than 200 or greater than 240; therefore, they cannot represent the total number of cookies on 4 trays. 
 

Question 127 127 of 569 selected Linear Equations In 1 Variable H

How many solutions does the equation 10(15x-9)=-15(6-10x) have?

  1. Exactly one

  2. Exactly two

  3. Infinitely many

  4. Zero

Show Answer Correct Answer: C

Choice C is correct. Applying the distributive property to each side of the given equation yields 150x-90=-90+150x. Applying the commutative property of addition to the right-hand side of this equation yields 150x-90=150x-90. Since the two sides of the equation are equivalent, this equation is true for any value of x . Therefore, the given equation has infinitely many solutions.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 128 128 of 569 selected Systems Of 2 Linear Equations In 2 Variables M

y=-2x

3x+y=40

The solution to the given system of equations is (x,y). What is the value of x?

Show Answer Correct Answer: 40

The correct answer is 40 . It’s given in the first equation of the system that y=-2x. Substituting - 2 x for y in the second equation of the system yields 3x+(-2x)=40. Combining like terms on the left-hand side of this equation yields x = 40 . Therefore, the value of x is 40 .

Question 129 129 of 569 selected Linear Functions E

Hydrogen is placed inside a container and kept at a constant pressure. The graph shows the estimated volume y , in liters, of the hydrogen when its temperature is x kelvins.

  • The line slants gradually up from left to right.
  • The line begins at the approximate point (90 comma 1.3).
  • The line passes through the following approximate points:
    • (90 comma 1.3)
    • (500 comma 7)

What is the estimated volume, in liters, of the hydrogen when its temperature is 500 kelvins?

  1. 0

  2. 7500

  3. 7

  4. 5007

Show Answer Correct Answer: C

Choice C is correct. For the graph shown, the x-axis represents temperature, in kelvins, and the y-axis represents volume, in liters. Therefore, the estimated volume, in liters, of the hydrogen when its temperature is 500 kelvins is represented by the y-coordinate of the point on the graph that has an x-coordinate of 500 . The point on the graph with an x-coordinate of 500 has a y-coordinate of 7 . Therefore, the estimated volume, in liters, of the hydrogen when its temperature is 500 kelvins is 7 .

Choice A is incorrect and may result from conceptual errors.

Choice B is incorrect and may result from conceptual errors.

Choice D is incorrect and may result from conceptual errors.

Question 130 130 of 569 selected Systems Of 2 Linear Equations In 2 Variables E

x + y = 18

5 y = x

What is the solution (x,y) to the given system of equations?

  1. (15,3)

  2. (16,2)

  3. (17,1)

  4. (18,0)

Show Answer Correct Answer: A

Choice A is correct. The second equation in the given system defines the value of x as 5 y . Substituting 5 y for x into the first equation yields 5y+y=18 or 6y=18. Dividing each side of this equation by 6 yields y = 3 . Substituting 3 for y in the second equation yields 5(3)=x or x=15. Therefore, the solution (x,y) to the given system of equations is (15,3)

Choice B is incorrect. Substituting 16 for x and 2 for y in the second equation yields 5(2)=16, which is not true. Therefore, (16,2) is not a solution to the given system of equations.

Choice C is incorrect. Substituting 17 for x and 1 for y in the second equation yields 5(1)=17, which is not true. Therefore, (17,1) is not a solution to the given system of equations.

Choice D is incorrect. Substituting 18 for x and 0 for y in the second equation yields 5(0)=18, which is not true. Therefore, (18,0) is not a solution to the given system of equations.

Question 131 131 of 569 selected Linear Functions E

The line graphed in the xy-plane below models the total cost, in dollars, for a cab ride, y, in a certain city during nonpeak hours based on the number of miles traveled, x.

The figure presents the graph of a line in the x y plane titled “Total Cost for a Cab Ride.” The x axis is labeled “Distance traveled, in miles,” and the integers 0, 5, and 10 are indicated. There are vertical gridlines at each integer from 1 through 10. The y axis is labeled “Cost, in dollars,” and the integers 0, 5, 10, and 15 are indicated. There are horizontal gridlines at each integer from 1 through 15. The line begins on the y axis at the point with coordinates 0 comma 3, and slants upward and to the right. It passes through the point with coordinates 5 comma 13.

According to the graph, what is the cost for each additional mile traveled, in dollars, of a cab ride?

  1. $2.00

  2. $2.60

  3. $3.00

  4. $5.00

Show Answer Correct Answer: A

Choice A is correct. The cost of each additional mile traveled is represented by the slope of the given line. The slope of the line can be calculated by identifying two points on the line and then calculating the ratio of the change in y to the change in x between the two points. Using the points with coordinates 1 comma 5 and 2 comma 7, the slope is equal to the fraction with numerator 7 minus 5, and denominator 2 minus 1, end fraction, or 2. Therefore, the cost for each additional mile traveled of the cab ride is $2.00.

Choice B is incorrect and may result from calculating the slope of the line that passes through the points with coordinates 5 comma 13 and 0 comma 0. However, the point with coordinates 0 comma 0 does not lie on the line shown. Choice C is incorrect. This is the y-coordinate of the y-intercept of the graph and represents the flat fee for a cab ride before the charge for any miles traveled is added. Choice D is incorrect. This value represents the total cost of a 1-mile cab ride.

Question 132 132 of 569 selected Linear Equations In 2 Variables H

In the xy-plane, line k intersects the y-axis at the point with coordinates 0 comma negative 6 and passes through the point with coordinates 2 comma 2. If the point with coordinates 20 comma w lies on line k, what is the value of w ?

Show Answer

The correct answer is 74. The y-intercept of a line in the xy-plane is the ordered pair x comma y of the point of intersection of the line with the y-axis. Since line k intersects the y-axis at the point with coordinates 0 comma negative 6, it follows that the point with coordinates 0 comma negative 6 is the y-intercept of this line. An equation of any line in the xy-plane can be written in the form y equals, m x plus b, where m is the slope of the line and b is the y-coordinate of the y-intercept. Therefore, the equation of line k can be written as y equals, m x plus negative 6, or y equals, m x minus 6. The value of m can be found by substituting the x- and y-coordinates from a point on the line, such as the point with coordinates 2 comma 2, for x and y, respectively. This results in 2 equals, 2 m minus 6. Solving this equation for m gives m equals 4. Therefore, an equation of line k is y equals, 4 x minus 6. The value of w can be found by substituting the x-coordinate, 20, for x in the equation of line k and solving this equation for y. This gives y equals, 4 times 20, minus 6, or y equals 74. Since w is the y-coordinate of this point, w equals 74.

Question 133 133 of 569 selected Linear Inequalities In 1 Or 2 Variables H

yx+7

y-2x-1

Which point (x,y) is a solution to the given system of inequalities in the xy-plane?

  1. (-14,0)

  2. (0,-14)

  3. (0,14)

  4. (14,0)

Show Answer Correct Answer: D

Choice D is correct. A point (x,y) is a solution to a system of inequalities in the xy-plane if substituting the x-coordinate and the y-coordinate of the point for x and y , respectively, in each inequality makes both of the inequalities true. Substituting the x-coordinate and the y-coordinate of choice D, 14 and 0 , for x and y , respectively, in the first inequality in the given system, yx+7, yields 014+7, or 021, which is true. Substituting 14 for x and 0 for y in the second inequality in the given system, y-2x-1, yields 0-2(14)-1, or 0-29, which is true. Therefore, the point (14,0) is a solution to the given system of inequalities in the xy-plane.

Choice A is incorrect. Substituting -14 for x and 0 for y in the inequality yx+7 yields 0-14+7, or 0-7, which is not true.

Choice B is incorrect. Substituting 0 for x and -14 for y in the inequality y-2x-1 yields -14-2(0)-1, or -14-1, which is not true.

Choice C is incorrect. Substituting 0 for x and 14 for y in the inequality yx+7 yields 140+7, or 147, which is not true.

Question 134 134 of 569 selected Linear Functions M

A team of workers has been moving cargo off of a ship. The equation below models the approximate number of tons of cargo, y, that remains to be moved x hours after the team started working.

y equals, 120 minus 25 x

The graph of this equation in the xy-plane is a line. What is the best interpretation of the x-intercept in this context?

  1. The team will have moved all the cargo in about 4.8 hours.

  2. The team has been moving about 4.8 tons of cargo per hour.

  3. The team has been moving about 25 tons of cargo per hour.

  4. The team started with 120 tons of cargo to move.

Show Answer Correct Answer: A

Choice A is correct. The x-intercept of the line with equation = 120 – 25x can be found by substituting 0 for y and finding the value of x. When y = 0, x = 4.8, so the x-intercept is at (4.8, 0). Since y represents the number of tons of cargo remaining to be moved x hours after the team started working, it follows that the x-intercept refers to the team having no cargo remaining to be moved after 4.8 hours. In other words, the team will have moved all of the cargo after about 4.8 hours.

Choice B is incorrect and may result from incorrectly interpreting the value 4.8. Choices C and D are incorrect and may result from misunderstanding the x-intercept. These statements are accurate but not directly relevant to the x-intercept.

 

Question 135 135 of 569 selected Systems Of 2 Linear Equations In 2 Variables M

y = 2 x + 10

y = 2 x - 1

At how many points do the graphs of the given equations intersect in the xy-plane?

  1. Zero

  2. Exactly one

  3. Exactly two

  4. Infinitely many

Show Answer Correct Answer: A

Choice A is correct. A system of two linear equations in two variables, x and y , has zero points of intersection if the lines represented by the equations in the xy-plane are distinct and parallel. The graphs of two lines in the xy-plane represented by equations in slope-intercept form, y = m x + b , are distinct if the y-coordinates of their y-intercepts, b , are different and are parallel if their slopes, m , are the same. For the two equations in the given system, y = 2 x + 10 and y = 2 x - 1 , the values of b are 10 and -1 , respectively, and the values of m are both 2 . Since the values of b are different, the graphs of these lines have different y-coordinates of the y-intercept and are distinct. Since the values of m are the same, the graphs of these lines have the same slope and are parallel. Therefore, the graphs of the given equations are lines that intersect at zero points in the xy-plane.

Choice B is incorrect. The graphs of a system of two linear equations have exactly one point of intersection if the lines represented by the equations have different slopes. Since the given equations represent lines with the same slope, there is not exactly one intersection point.

Choice C is incorrect. The graphs of a system of two linear equations can never have exactly two intersection points.

Choice D is incorrect. The graphs of a system of two linear equations have infinitely many intersection points when the lines represented by the equations have the same slope and the same y-coordinate of the y-intercept. Since the given equations represent lines with different y-coordinates of their y-intercepts, there are not infinitely many intersection points.

Question 136 136 of 569 selected Linear Equations In 2 Variables E
The figure presents the graph of a line in the x y plane. The number 1 is indicated on both axes. The line passes through the point with coordinates negative 4 comma 0, and the point with coordinates 0 comma negative 1.

Which of the following is an equation of the graph shown in the xy-plane above?

  1.  y equals, negative one fourth x, minus 1

  2. y equals, negative x, minus 4

  3. y equals, negative x, minus one fourth

  4. y equals, negative 4 x, minus 1

Show Answer Correct Answer: A

Choice A is correct. The slope of the line can be found by choosing any two points on the line, such as (4, –2) and (0, –1). Subtracting the y-values results in –2 – (–1) = –1, the change in y. Subtracting the x-values results in 4 – 0 = 4, the change in x. Dividing the change in y by the change in x yields negative 1, divided by 4, equals negative one fourth, the slope. The line intersects the y-axis at (0, –1), so –1 is the y-coordinate of the y-intercept. This information can be expressed in slope-intercept form as the equation y equals, negative one fourth, x, minus 1.

Choice B is incorrect and may result from incorrectly calculating the slope and then misidentifying the slope as the y-intercept. Choice C is incorrect and may result from misidentifying the slope as the y-intercept. Choice D is incorrect and may result from incorrectly calculating the slope.

 

Question 137 137 of 569 selected Systems Of 2 Linear Equations In 2 Variables H

4 x - 6 y =10y+2

ty=12+2x

In the given system of equations, t is a constant. If the system has no solution, what is the value of t ?

Show Answer Correct Answer: 8

The correct answer is 8 . The given system of equations can be solved using the elimination method. Multiplying both sides of the second equation in the given system by -2 yields -2ty=-1-4x, or -1-4x=-2ty. Adding this equation to the first equation in the given system, 4x-6y=10y+2, yields (4x-6y)+(-1-4x)=(10y+2)+(-2ty), or -1-6y=10y-2ty+2. Subtracting 10 y from both sides of this equation yields (-1-6y)-(10y)=(10y-2ty+2)-(10y), or -1-16y=-2ty+2. If the given system has no solution, then the equation -1-16y=-2ty+2 has no solution. If this equation has no solution, the coefficients of y on each side of the equation, -16 and -2t, must be equal, which yields the equation -16=-2t. Dividing both sides of this equation by -2 yields 8=t. Thus, if the system has no solution, the value of t is 8 .

Alternate approach: A system of two linear equations in two variables, x and y , has no solution if the lines represented by the equations in the xy-plane are parallel and distinct. Lines represented by equations in the form Ax+By=C, where A , B , and C are constant terms, are parallel if the ratio of the x-coefficients is equal to the ratio of the y-coefficients, and distinct if the ratio of the x-coefficients are not equal to the ratio of the constant terms. Subtracting 10y from both sides of the first equation in the given system yields (4x-6y)-(10y)=(10y+2)-(10y), or 4x-16y=2. Subtracting 2x from both sides of the second equation in the given system yields (ty)-(2x)=(12+2x)-(2x), or -2x+ty=12. The ratio of the x-coefficients for these equations is -24, or -12. The ratio of the y-coefficients for these equations is -t16. The ratio of the constant terms for these equations is 122, or 14. Since the ratio of the x-coefficients, -12, is not equal to the ratio of the constants, 14, the lines represented by the equations are distinct. Setting the ratio of the x-coefficients equal to the ratio of the y-coefficients yields -12=-t16. Multiplying both sides of this equation by -16 yields (-12)(-16)=(-t16)(-16), or t=8. Therefore, when t=8, the lines represented by these equations are parallel. Thus, if the system has no solution, the value of t is 8 .

Question 138 138 of 569 selected Linear Equations In 2 Variables H

35x+34y=7

Which table gives three values of x and their corresponding values of y for the given equation?

Show Answer Correct Answer: D

Choice D is correct. Each of the tables gives the same three values of x : 1 , 2 , and 4 . Substituting 1 for x in the given equation yields (35)(1)+34y=7, or 35+34y=355. Subtracting 3 5 from both sides of this equation yields 34y=325. Multiplying both sides of this equation by 4 3 yields y = 128 15 . Therefore, when x = 1 , the corresponding value of y for the given equation is 128 15 . Substituting 2 for x in the given equation yields (35)(2)+34y=7, or 65+34y=355. Subtracting 6 5 from both sides of this equation yields 34y=295. Multiplying both sides of this equation by 4 3 yields y = 116 15 . Therefore, when x = 2 , the corresponding value of y for the given equation is 116 15 . Substituting 4 for x in the given equation yields (35)(4)+34y=7, or 125+34y=355. Subtracting 12 5 from both sides of this equation yields 34y=235. Multiplying both sides of this equation by 4 3 yields y = 92 15 . Therefore, when x = 4 , the corresponding value of y for the given equation is 92 15 . The table in choice D gives x-values of 1 , 2 , and 4 and corresponding y-values of 128 15 , 116 15 , and 92 15 , respectively. Therefore, the table in choice D gives three values of x and their corresponding values of y for the given equation.

Choice A is incorrect. This table gives three values of x and their corresponding values of y for the equation 35x+34+y=7.

Choice B is incorrect. This table gives three values of x and their corresponding values of y for the equation 35x+y=10.

Choice C is incorrect. This table gives three values of x and their corresponding values of y for the equation 35x+34y=8.

Question 139 139 of 569 selected Linear Functions M

In the xy-plane, line k has a slope of 5 and a y-intercept of (0,-35). What is the x-coordinate of the x-intercept of line k ?

Show Answer Correct Answer: 7

The correct answer is 7. An equation of a line in the xy-plane can be written in the form y=mx+b, where m is the slope of the line and (0,b) is the y-intercept of the line. It’s given that line k has a slope of 5 and a y-intercept of (0,35). Therefore, m=5 and b=35. Substituting 5 for m and 35 for b in the equation y=mx+b yields y=5x35. The x-intercept of a line in the xy-plane is the point where the line intersects the x-axis, which is a point with a y-coordinate of 0. Substituting 0 for y in the equation y=5x35 yields 0=5x35. Adding 35 to both sides of this equation yields 35=5x. Dividing both sides of this equation by 5 yields 7=x. Therefore, the x-coordinate of the x-intercept of line k is 7.

Question 140 140 of 569 selected Linear Functions E

A contract for a certain service requires a onetime activation cost of $35 and a monthly cost of $23. Which equation represents this situation, where c is the total cost, in dollars, of this service contract for t months?

  1. c=t23+35

  2. c=t35+23

  3. c=23t+35

  4. c=35t+23

Show Answer Correct Answer: C

Choice C is correct. It's given that this service contract requires a monthly cost of $23. A monthly cost of $23 for t months results in a cost of $23t. It's also given that this service contract requires a onetime activation cost of $35. Adding the onetime activation cost to the monthly cost of the service contract for t months yields the total cost c , in dollars, of this service contract for t months. Therefore, this situation can be represented by the equation c=23t+35.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 141 141 of 569 selected Linear Equations In 2 Variables E

An employee at a restaurant prepares sandwiches and salads. It takes the employee 1.5 minutes to prepare a sandwich and 1.9 minutes to prepare a salad. The employee spends a total of 46.1 minutes preparing x sandwiches and y salads. Which equation represents this situation?

  1. 1.9x+1.5y=46.1

  2. 1.5x+1.9y=46.1

  3. x+y=46.1

  4. 30.7x+24.3y=46.1

Show Answer Correct Answer: B

Choice B is correct. It’s given that the employee takes 1.5 minutes to prepare a sandwich. Multiplying 1.5 by the number of sandwiches, x, yields 1.5x, the amount of time the employee spends preparing x sandwiches. It’s also given that the employee takes 1.9 minutes to prepare a salad. Multiplying 1.9 by the number of salads, y, yields 1.9y, the amount of time the employee spends preparing y salads. It follows that the total amount of time, in minutes, the employee spends preparing x sandwiches and y salads is 1.5x+1.9y. It's given that the employee spends a total of 46.1 minutes preparing x sandwiches and y salads. Thus, the equation 1.5x+1.9y=46.1 represents this situation.

Choice A is incorrect. This equation represents a situation where it takes the employee 1.9 minutes, rather than 1.5 minutes, to prepare a sandwich and 1.5 minutes, rather than 1.9 minutes, to prepare a salad.

Choice C is incorrect. This equation represents a situation where it takes the employee 1 minute, rather than 1.5 minutes, to prepare a sandwich and 1 minute, rather than 1.9 minutes, to prepare a salad.

Choice D is incorrect. This equation represents a situation where it takes the employee 30.7 minutes, rather than 1.5 minutes, to prepare a sandwich and 24.3 minutes, rather than 1.9 minutes, to prepare a salad.

Question 142 142 of 569 selected Linear Functions H

F(x)=95(x-273.15)+32

The function F gives the temperature, in degrees Fahrenheit, that corresponds to a temperature of x kelvins. If a temperature increased by 9.10 kelvins, by how much did the temperature increase, in degrees Fahrenheit?

  1. 16.38

  2. 48.38

  3. 475.29

  4. 507.29

Show Answer Correct Answer: A

Choice A is correct. It’s given that the function F(x)=95(x-273.15)+32 gives the temperature, in degrees Fahrenheit, that corresponds to a temperature of x kelvins. A temperature that increased by 9.10 kelvins means that the value of x increased by 9.10 kelvins. It follows that an increase in x by 9.10 increases F(x) by 95(9.10), or 16.38. Therefore, if a temperature increased by 9.10 kelvins, the temperature increased by 16.38 degrees Fahrenheit. 

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 143 143 of 569 selected Linear Equations In 1 Variable E

 

the fraction 4 x over 5, equals 20

In the equation above, what is the value of x ?

 

  1. 25

  2. 24

  3. 16

  4. 15

Show Answer Correct Answer: A

Choice A is correct. Multiplying both sides of the equation by 5 results in 4 x equals 100. Dividing both sides of the resulting equation by 4 results in x equals 25.

Choice B is incorrect and may result from adding 20 and 4. Choice C is incorrect and may result from dividing 20 by 5 and then multiplying the result by 4. Choice D is incorrect and may result from subtracting 5 from 20.

 

Question 144 144 of 569 selected Linear Functions M

The pressure exerted on a scuba diver at sea level is 14.70 pounds per square inch (psi). For each foot the scuba diver descends below sea level, the pressure exerted on the scuba diver increases by 0.44 psi. What is the total pressure, in psi, exerted on the scuba diver at 105 feet below sea level?

  1. 60.90

  2. 31.50

  3. 14.70

  4. 0.44

Show Answer Correct Answer: A

Choice A is correct. It's given that the pressure exerted on a scuba diver at sea level is 14.70 pounds per square inch (psi). It's also given that for each foot the scuba diver descends below sea level, the pressure exerted on the scuba diver increases by 0.44 psi. The total pressure, in psi, exerted on the scuba diver at x feet below sea level can be represented by the expression 0.44x+14.70. Substituting 105 for x in this expression yields 0.44(105)+14.70, or 60.90. Therefore, the total pressure exerted on the scuba diver at 105 feet below sea level is 60.90 psi.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect. This is the pressure, in psi, exerted on the scuba diver at sea level, not at 105 feet below sea level.

Choice D is incorrect. This is the rate by which the pressure, in psi, exerted on the scuba diver increases for each foot the scuba diver descends below sea level.

Question 145 145 of 569 selected Linear Inequalities In 1 Or 2 Variables M

A truck can haul a maximum weight of 5,630 pounds. During one trip, the truck will be used to haul a 190 -pound piece of equipment as well as several crates. Some of these crates weigh 25 pounds each and the others weigh 62 pounds each. Which inequality represents the possible combinations of the number of 25 -pound crates, x , and the number of 62 -pound crates, y , the truck can haul during one trip if only the piece of equipment and the crates are being hauled?

  1. 25x+62y5,440

  2. 25x+62y5,440

  3. 62x+25y5,630

  4. 62x+25y5,630

Show Answer Correct Answer: A

Choice A is correct. It's given that a truck can haul a maximum of 5,630 pounds. It's also given that during one trip, the truck will be used to haul a 190 -pound piece of equipment as well as several crates. It follows that the truck can haul at most 5,630-190, or 5,440 , pounds of crates. Since x represents the number of 25 -pound crates, the expression 25 x represents the weight of the 25 -pound crates. Since y represents the number of 62 -pound crates, 62 y represents the weight of the 62 -pound crates. Therefore, 25 x + 62 y represents the total weight of the crates the truck can haul. Since the truck can haul at most 5,440 pounds of crates, the total weight of the crates must be less than or equal to 5,440 pounds, or 25x+62y5,440.

Choice B is incorrect. This represents the possible combinations of the number of 25 -pound crates, x , and the number of 62 -pound crates, y , the truck can haul during one trip if it can haul a minimum, not a maximum, of 5,630 pounds.

Choice C is incorrect. This represents the possible combinations of the number of 62 -pound crates, x , and the number of 25 -pound crates, y , the truck can haul during one trip if only crates are being hauled.

Choice D is incorrect. This represents the possible combinations of the number of 62 -pound crates, x , and the number of 25 -pound crates, y , the truck can haul during one trip if it can haul a minimum, not a maximum, weight of 5,630 pounds and only crates are being hauled.

Question 146 146 of 569 selected Systems Of 2 Linear Equations In 2 Variables M

Equation 1: x plus 5 y, equals 5. Equation 2: 2 x minus y, equals negative 4

Which of the following graphs in the xy-plane could be used to solve the system of equations above?

  1. The answer choice presents the graph of two lines in the x y plane. The numbers negative 4 and 4 are indicated on each axis. One line slants upward and to the right. It crosses the x axis at negative 5, and crosses the y axis at 1. The other line slants downward and to the right. It crosses the y axis at 4, and crosses the x axis at 2. The two lines intersect above the x axis and to the right of the y axis.

  2. The answer choice presents the graph of two lines in the x y plane. The numbers negative 4 and 4 are indicated on each axis. One line slants upward and to the right. It crosses the y axis at negative 4, and crosses the x axis at 4. The other line slants downward and to the right. It crosses the y axis at 1, and crosses the x axis at 1. The two lines intersect below the x axis and to the right of the y axis.

  3. The answer choice presents the graph of two lines in the x y plane. The numbers negative 4 and 4 are indicated on each axis. One line slants upward and to the right. It crosses the x axis at negative 2, and crosses the y axis at 4. The other line slants downward and to the right. It crosses the y axis at 1, and crosses the x axis at 5. The two lines intersect above the x axis and to the left of the y axis.

  4. The answer choice presents the graph of two lines in the x y plane. The numbers negative 4 and 4 are indicated on each axis. One line slants upward and to the right. It crosses the x axis at negative 2, and crosses the y axis at 4. The other line slants upward and to the right. It crosses the y axis at negative 1, and crosses the x axis at 5. The two lines intersect below the x axis and to the left of the y axis.

Show Answer Correct Answer: C

Choice C is correct. The graph of a system of equations is the graph that shows the lines represented by each of the equations in the system. The x-intercept of the graph of each given equation can be found by substituting 0 for y in each equation: x plus, 5 times 0, equals 5, or x equals 5, and 2 x minus 0, equals negative 4, or x equals negative 2. The y-intercept of the graph of each equation can be found by substituting 0 for x in each equation: 0 plus 5 y, equals 5, or y equals 1, and 2 times 0, minus y, equals negative 4 or y equals 4. Using these x- and y- intercept values, the line that has equation x plus 5 y, equals 5 passes through the points with coordinates 0 comma 1 and 5 comma 0, and the line that has equation 2 x minus y, equals negative 4 passes through the points with coordinates 0 comma 4 and negative 2 comma 0. Only the lines in choice C pass through these points and can be used to solve the given system of equations.

Choices A, B, and D are incorrect. In choices A and B, neither line passes through the points with coordinates 0 comma 1 and 5 comma 0 or the points with coordinates 0 comma 4 and negative 2 comma 0. In choice D, although one line passes through the points with coordinates 0 comma 4 and negative 2 comma 0 the other line doesn’t pass through the points with coordinates 0 comma 1 and 5 comma 0.

 

Question 147 147 of 569 selected Linear Equations In 1 Variable H

3(kx+13)= 48 17 x + 36

In the given equation, k is a constant. The equation has no solution. What is the value of k ?

Show Answer Correct Answer: .9411, .9412, 16/17

The correct answer is 1617. It's given that the equation 3(kx+13)=4817x+36 has no solution. A linear equation in the form ax+b=cx+d, where a , b , c , and d are constants, has no solution only when the coefficients of x on each side of the equation are equal and the constant terms aren't equal. Dividing both sides of the given equation by 3 yields kx+13=4851x+363, or kx+13=1617x+12. Since the coefficients of x on each side of the equation must be equal, it follows that the value of k is 16 17 . Note that 16/17, .9411, .9412, and 0.941 are examples of ways to enter a correct answer.

Question 148 148 of 569 selected Linear Functions M

The function f is defined by f(x)=7x-84. What is the x-intercept of the graph of y=f(x) in the xy-plane?

  1. (-12,0)

  2. (-7,0)

  3. (7,0)

  4. (12,0)

Show Answer Correct Answer: D

Choice D is correct. The given function f is a linear function. Therefore, the graph of y=f(x) in the xy-plane has one x-intercept at the point (k,0), where k is a constant. Substituting 0 for f(x) and k for x in the given function yields 0=7k-84. Adding 84 to both sides of this equation yields 84=7k. Dividing both sides of this equation by 7 yields 12=k. Therefore, the x-intercept of the graph of y=f(x) in the xy-plane is (12,0).

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Question 149 149 of 569 selected Linear Functions M
Distance (kilometers) Average time (minutes)
0.32 8
0.56 14
0.68 17

The table gives the average time t , in minutes, it takes Carly to travel a certain distance d , in kilometers. Which equation could represent this linear relationship?

  1. t = 4 d

  2. t=125d

  3. t = 25 d

  4. t=14d

Show Answer Correct Answer: C

Choice C is correct. The average time t, in minutes, it takes Carly to travel a certain distance d, in kilometers, is given in the table. This linear relationship can be represented by an equation in the form t=ad+b, where a and b are constants. The table shows that it takes Carly an average time of 8 minutes to travel 0.32 kilometers. Substituting 8 for t and 0.32 for d in the equation t=ad+b yields 8=0.32a+b. Subtracting 0.32a from both sides of this equation yields 8-0.32a=b. The table also shows that it takes Carly an average time of 14 minutes to travel 0.56 kilometers. Substituting 14 for t and 0.56 for d in the equation t=ad+b yields 14=0.56a+b. Subtracting 0.56a from both sides of this equation yields 14-0.56a=b. Substituting 8-0.32a for b in this equation yields 14-0.56a=8-0.32a. Subtracting 8 from both sides of this equation yields 6-0.56a=-0.32a. Adding 0.56a to both sides of this equation yields 6=0.24a. Dividing both sides of this equation by 0.24 yields 25=a. Substituting 25 for a in the equation 8=0.32a+b yields 8=0.32(25)+b, or 8=8+b. Subtracting 8 from both sides of this equation yields 0=b. Substituting 25 for a and 0 for b in the equation t=ad+b yields t=25d+0, or t=25d.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 150 150 of 569 selected Linear Functions M
x 10 15 20 25
f(x) 82 137 192 247

The table shows four values of x and their corresponding values of f(x). There is a linear relationship between x  and f(x) that is defined by the equation f(x)=mx-28, where m is a constant. What is the value of m ?

Show Answer Correct Answer: 11

The correct answer is 11 . It's given that f(x) is defined by the equation f(x)=mx-28, where m is a constant. It's also given in the table that when x = 10 , f(x)=82. Substituting 10 for x and 82 for f(x) in the equation f(x)=mx-28 yields, 82=m(10)-28. Adding 28 to both sides of this equation yields 110 = 10 m . Dividing both sides of this equation by 10 yields 11 = m . Therefore, the value of m is 11 .

Question 151 151 of 569 selected Linear Inequalities In 1 Or 2 Variables E

Julissa needs at least 100 hours of flight time to get her private pilot certification. If Julissa already has 86 hours of flight time, what is the minimum number of additional hours of flight time Julissa needs to get her private pilot certification?

  1. 14

  2. 76

  3. 86

  4. 186

Show Answer Correct Answer: A

Choice A is correct. It's given that Julissa already has 86 hours of flight time. Let x represent the number of additional hours of flight time Julissa needs to get her private pilot certification. After completing x hours of flight time, Julissa will have completed a total of 86+x hours of flight time. It's given that Julissa needs at least 100 hours of flight time to get her private pilot certification. Therefore, 86+x100. Subtracting 86 from both sides of this inequality yields x14. Thus, 14 is the minimum number of additional hours of flight time Julissa needs to get her private pilot certification.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect. This is the number of hours of flight time Julissa already has, rather than the minimum number of additional hours of flight time Julissa needs.

Choice D is incorrect. This is the number of hours of flight time Julissa will have if she completes 100 more hours of flight time, rather than the minimum number of additional hours of flight time Julissa needs.

Question 152 152 of 569 selected Linear Functions H

  • The line slants sharply down from left to right.
  • The line passes through the following points:
    • (negative 1 comma 7)
    • (0 comma 3)

The graph of the linear function y=f(x)+19 is shown. If c and d are positive constants, which equation could define f ?

  1. f(x)= - d - c x

  2. f(x)= d - c x

  3. f(x)= - d + c x

  4. f(x)= d + c x

Show Answer Correct Answer: A

Choice A is correct. It’s given that the graph of the linear function y=f(x)+19 is shown. This means that the graph of y=f(x)+19 can be translated down 19 units to create the graph of y=f(x) and the y-coordinate of every point on the graph of y=f(x)+19 can be decreased by 19 to find the resulting point on the graph of y=f(x). The y-intercept of the graph of y=f(x)+19 is (0,3). Translating the graph of y=f(x)+19 down 19 units results in a y-intercept of the graph of y=f(x) at the point (0,3-19), or (0,-16). The graph of y=f(x)+19 slants down from left to right, so the slope of the graph is negative. The translation of a linear graph changes its position, but does not change its slope. It follows that the slope of the graph of y=f(x) is also negative. The equation of a linear function f can be written in the form f(x)=b+mx, where b is the y-coordinate of the y-intercept and m is the slope of the graph of y=f(x). It's given that c and d are positive constants. Since the y-coordinate of the y-intercept and the slope of the graph of y=f(x) are both negative, it follows that f(x)=-d-cx could define f .

Choice B is incorrect. This could define a linear function where its graph has a positive, not negative, y-intercept.

Choice C is incorrect. This could define a linear function where its graph has a positive, not negative, slope.

Choice D is incorrect. This could define a linear function where its graph has a positive, not negative, y-intercept and a positive, not negative, slope.

Question 153 153 of 569 selected Systems Of 2 Linear Equations In 2 Variables E

x = 10

y = x + 21

The solution to the given system of equations is (x,y). What is the value of y ?

  1. 2.1

  2. 10

  3. 21

  4. 31

Show Answer Correct Answer: D

Choice D is correct. It's given by the first equation in the given system of equations that x = 10 . Substituting 10 for x in the second equation in the given system yields y=10+21, or y = 31 . Therefore, the value of y is 31 .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect. This is the value of x , not the value of y .

Choice C is incorrect and may result from conceptual or calculation errors.

Question 154 154 of 569 selected Systems Of 2 Linear Equations In 2 Variables M

y=-19x

y=12x

The solution to the given system of equations is (x,y). What is the value of x ?

  1. -9

  2. -7

  3. 0

  4. 2

Show Answer Correct Answer: C

Choice C is correct. It's given by the first equation in the system that y=-19x. Substituting -19x for y in the second equation in the system yields -19x=12x. Multiplying the left-hand side of this equation by 22 and the right-hand side by 99 yields -218x=918x. Adding 218x to both sides of this equation yields 0=1118x. Multiplying both sides of this equation by 1811 yields x = 0 .

Choice A is incorrect and may result from conceptual or calculation errors. 

Choice B is incorrect and may result from conceptual or calculation errors. 

Choice D is incorrect and may result from conceptual or calculation errors. 

Question 155 155 of 569 selected Linear Inequalities In 1 Or 2 Variables H

A salesperson’s total earnings consist of a base salary of x dollars per year, plus commission earnings of 11 % of the total sales the salesperson makes during the year. This year, the salesperson has a goal for the total earnings to be at least 3 times and at most 4 times the base salary. Which of the following inequalities represents all possible values of total sales s , in dollars, the salesperson can make this year in order to meet that goal? 

  1. 2 x s 3 x

  2. 20.11xs30.11x

  3. 3 x s 4 x

  4. 30.11xs40.11x

Show Answer Correct Answer: B

Choice B is correct. It’s given that a salesperson's total earnings consist of a base salary of x dollars per year plus commission earnings of 11% of the total sales the salesperson makes during the year. If the salesperson makes s dollars in total sales this year, the salesperson’s total earnings can be represented by the expression x+0.11s. It’s also given that the salesperson has a goal for the total earnings to be at least 3 times and at most 4 times the base salary, which can be represented by the expressions 3 x and 4 x , respectively. Therefore, this situation can be represented by the inequality 3xx+0.11s4x. Subtracting x from each part of this inequality yields 2x0.11s3x. Dividing each part of this inequality by 0.11 yields 20.11xs30.11x. Therefore, the inequality 20.11xs30.11x represents all possible values of total sales s , in dollars, the salesperson can make this year in order to meet their goal.

Choice A is incorrect. This inequality represents a situation in which the total sales, rather than the total earnings, are at least 2 times and at most 3 times, rather than at least 3 times and at most 4 times, the base salary.

Choice C is incorrect. This inequality represents a situation in which the total sales, rather than the total earnings, are at least 3 times and at most 4 times the base salary.

Choice D is incorrect. This inequality represents a situation in which the total earnings are at least 4 times and at most 5 times, rather than at least 3 times and at most 4 times, the base salary.

Question 156 156 of 569 selected Linear Functions H
x f(x)
-4 0
-195 1
-185 2

For the linear function f , the table shows three values of x and their corresponding values of f(x). If h(x)=f(x)-13, which equation defines h ?

  1. h(x)=5x-4

  2. h(x)=5x+7

  3. h(x)=5x+9

  4. h(x)=5x+20

Show Answer Correct Answer: B

Choice B is correct. An equation that defines a linear function f can be written in the form f(x)=mx+b, where m and b are constants. It's given in the table that when x = - 4 , f(x)=0. Substituting - 4 for x and 0 for f(x) in the equation f(x)=mx+b yields 0=m(-4)+b, or 0=-4m+b. Adding 4 m to both sides of this equation yields 4 m = b . Substituting 4 m for b in the equation f(x)=mx+b yields f(x)=mx+4m. It's also given in the table that when x = - 19 5 , f(x)=1. Substituting - 19 5 for x and 1 for f(x) in the equation f(x)=mx+4m yields 1=m(-195)+4m, or 1=15m. Multiplying both sides of this equation by 5 yields m = 5 . Substituting 5 for m in the equation f(x)=mx+4m yields f(x)=5x+4(5), or f(x)=5x+20. If h(x)=f(x)-13, substituting 5 x + 20 for f(x) in this equation yields h(x)=(5x+20)-13, or h(x)=5x+7.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect. This is an equation that defines the linear function f , not h .

Question 157 157 of 569 selected Linear Equations In 1 Variable E

Which of the following is equivalent to 4 x plus 6 equals 12 ?

  1. 2 x plus 4, equals 6

  2. x plus 3, equals 3

  3. 3 x plus 2, equals 4

  4. 2 x plus 3, equals 6

Show Answer Correct Answer: D

Choice D is correct. Dividing each side of the original equation by yields the fraction with numerator 4 x plus 6, and denominator 2, equals twelve halves, which simplifies to 2 x plus 3, equals 6.

Choice A is incorrect. Dividing each side of the original equation by gives 2 x plus 3, equals 6, which is not equivalent to 2 x plus 4, equals 6. Choice B is incorrect. Dividing each side of the original equation by  gives x plus three halves, equals 3, which is not equivalent to x plus 3, equals 3. Choice C is incorrect. Dividing each side of the original equation by gives four thirds x, plus 2, equals 4, which is not equivalent to 3 x plus 2, equals 4

 

Question 158 158 of 569 selected Linear Equations In 2 Variables M

What is the slope of the graph of 10 x - 5 y = -12 in the xy-plane?

  1. -2

  2. - 5 6

  3. 5 6

  4. 2

Show Answer Correct Answer: D

Choice D is correct. A linear equation can be written in the form y = m x + b , where m is the slope of the graph of the equation in the xy-plane and (0,b) is the y-intercept. Subtracting 10 x from each side of the given equation, 10 x - 5 y = - 12 , yields - 5 y = - 10 x - 12 . Dividing each side of this equation by - 5 yields y = 2 x + 12 5 . This equation is in the form y = m x + b , where m = 2 . Therefore, the slope of the graph of the given equation in the xy-plane is 2 .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Question 159 159 of 569 selected Linear Equations In 2 Variables H

In the xy-plane, line p has a slope of - 5 3 and an x-intercept of (-6,0). What is the y-coordinate of the y-intercept of line p ?

Show Answer Correct Answer: -10

The correct answer is -10 . A line in the xy-plane can be represented by the equation y=mx+b, where m is the slope of the line and b is the y-coordinate of the y-intercept. It's given that line p has a slope of -53. Therefore, m=-53. It's also given that line p has an x-intercept of (-6,0). Therefore, when x = -6 , y = 0 . Substituting -53 for m , -6 for x , and 0 for y in the equation y=mx+b yields 0=(-53)(-6)+b, which is equivalent to 0=10+b. Subtracting 10 from both sides of this equation yields -10 = b . Therefore, the y-coordinate of the y-intercept of line p is -10 .

Question 160 160 of 569 selected Linear Equations In 1 Variable E

One pound of grapes costs $2. At this rate, how many dollars will pounds of grapes cost?

  1. 2 c

  2. 2 plus c

  3. the fraction 2 over c

  4. the fraction c over 2

Show Answer Correct Answer: A

Choice A is correct. If one pound of grapes costs $2, two pounds of grapes will cost 2 times $2, three pounds of grapes will cost 3 times $2, and so on. Therefore, c pounds of grapes will cost c times $2, which is 2c dollars.

Choice B is incorrect and may result from incorrectly adding instead of multiplying. Choice C is incorrect and may result from assuming that c pounds cost $2, and then finding the cost per pound. Choice D is incorrect and could result from incorrectly assuming that 2 pounds cost $c, and then finding the cost per pound.

 

Question 161 161 of 569 selected Linear Functions M

The graph of the function f , where y=f(x), gives the total cost y , in dollars, for a certain video game system and x games. What is the best interpretation of the slope of the graph in this context?

  1. Each game costs $25 .

  2. The video game system costs $100 .

  3. The video game system costs $25 .

  4. Each game costs $100 .

Show Answer Correct Answer: A

Choice A is correct. The given graph is a line, and the slope of a line is defined as the change in the value of y for each increase in the value of x by 1 . It’s given that y represents the total cost, in dollars, and that x represents the number of games. Therefore, the change in the value of y for each increase in the value of x by 1 represents the change in total cost, in dollars, for each increase in the number of games by 1 . In other words, the slope represents the cost, in dollars, per game. The graph shows that when the value of x increases from 0 to 1 , the value of y increases from 100 to 125 . It follows that the slope is 25 , or the cost per game is $25. Thus, the best interpretation of the slope of the graph is that each game costs $25.

Choice B is incorrect. This is an interpretation of the y-intercept of the graph rather than the slope of the graph.

Choice C is incorrect. The slope of the graph is the cost per game, not the cost of the video game system.

Choice D is incorrect. Each game costs $25, not $100.

Question 162 162 of 569 selected Systems Of 2 Linear Equations In 2 Variables M

A bus traveled on the highway and on local roads to complete a trip of 160 miles. The trip took 4 hours. The bus traveled at an average speed of 55 miles per hour (mph) on the highway and an average speed of 25 mph on local roads. If x is the time, in hours, the bus traveled on the highway and y is the time, in hours, it traveled on local roads, which system of equations represents this situation?

  1. 55x+25y=4
    x+y=160

  2. 55x+25y=160
    x+y=4

  3. 25x+55y=4
    x+y=160

  4. 25x+55y=160
    x+y=4

Show Answer Correct Answer: B

Choice B is correct. If the bus traveled at an average speed of 55 miles per hour (mph) on the highway for x hours, then the bus traveled 55 x miles on the highway. If the bus traveled at an average speed of 25 mph on local roads for y hours, then the bus traveled 25 y miles on local roads. It's given that the trip was 160 miles. This can be represented by the equation 55x+25y=160. It's also given that the trip took 4 hours. This can be represented by the equation x + y = 4 . Therefore, the system consisting of the equations 55x+25y=160 and x + y = 4 represents this situation.

Choice A is incorrect. This system of equations represents a situation where the trip was 4 miles and took 160 hours.

Choice C is incorrect. This system of equations represents a situation where the trip was 4 miles and took 160 hours, and the bus traveled at an average speed of 25 mph on the highway and 55 mph on local roads.

Choice D is incorrect. This system of equations represents a situation where the bus traveled at an average speed of 25 mph on the highway and 55 mph on local roads.

Question 163 163 of 569 selected Linear Equations In 2 Variables M

On a 210-mile trip, Cameron drove at an average speed of 60 miles per hour for the first x hours. He then completed the trip, driving at an average speed of 50 miles per hour for the remaining y hours. If x equals 1, what is the value of y ?

Show Answer

The correct answer is 3. It’s given that Cameron drove 60 miles per hour for x hours; therefore, the distance driven at this speed can be represented by 60 x. He then drove 50 miles per hour for y hours; therefore, the distance driven at this speed can be represented by 50 y. Since Cameron drove 210 total miles, the equation 60 x plus 50 y, equals 210 represents this situation. If x equals 1, substitution yields 60 times 1, plus 50 y, equals 210, or 60 plus 50 y, equals 210. Subtracting 60 from both sides of this equation yields 50 y equals 150. Dividing both sides of this equation by 50 yields y equals 3..

Question 164 164 of 569 selected Linear Functions E

For the linear function f , the graph of y=f(x) in the xy-plane has a slope of 2 and has a y-intercept at (0,-5). Which equation defines f ?

  1. f(x)=12x-5

  2. f(x)=-12x-5

  3. f(x)=-2x-5

  4. f(x)=2x-5

Show Answer Correct Answer: D

Choice D is correct. An equation defining the linear function f can be written in the form f(x)=mx+b, where m is the slope and (0,b) is the y-intercept of the graph of y=f(x) in the xy-plane. It’s given that the graph of y=f(x) has a slope of 2 . Therefore, m=2. It’s also given that the graph of y=f(x) has a y-intercept at (0,-5). Therefore, b=-5. Substituting 2 for m and -5 for b in the equation f(x)=mx+b yields f(x)=2x-5. Thus, the equation that defines f is f(x)=2x-5.

Choice A is incorrect. For this function, the graph of y=f(x) in the xy-plane has a slope of 12, not 2 .

Choice B is incorrect. For this function, the graph of y=f(x) in the xy-plane has a slope of -12, not 2 .

Choice C is incorrect. For this function, the graph of y=f(x) in the xy-plane has a slope of -2 , not 2 .

Question 165 165 of 569 selected Linear Inequalities In 1 Or 2 Variables E

Ty set a goal to walk at least 24 kilometers every day to prepare for a multiday hike. On a certain day, Ty plans to walk at an average speed of 4 kilometers per hour. What is the minimum number of hours Ty must walk on that day to fulfill the daily goal?

  1. 4

  2. 6

  3. 20

  4. 24

Show Answer Correct Answer: B

Choice B is correct. It's given that Ty plans to walk at an average speed of 4 kilometers per hour. The number of kilometers Ty will walk is determined by the expression 4 s , where s is the number of hours Ty walks. The given goal of at least 24 kilometers means that the inequality 4s24 represents the situation. Dividing both sides of this inequality by 4 gives s6 , which corresponds to a minimum of 6 hours Ty must walk. 

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 166 166 of 569 selected Systems Of 2 Linear Equations In 2 Variables E

Equation 1: y equals 2 x plus 3. Equation 2: x equals 1.

What is the solution the ordered pair, x comma y, to the given system of equations?

  1. 1 comma 2

  2. 1 comma 5

  3. 2 comma 3

  4. 2 comma 7

Show Answer Correct Answer: B

Choice B is correct. Since it’s given that x equals 1, substituting 1 for x in the first equation yields y equals, 2 times 1, plus 3. Simplifying the right-hand side of this equation yields y equals, 2 plus 3, or y equals 5. Therefore, the ordered pair 1 comma 5 is a solution to the given system of equations.

Choice A is incorrect and may result from a calculation error when substituting 1 for x in the first equation. Choices C and D are incorrect. Because it’s given that x equals 1, x cannot equal 2 as stated in these ordered pairs.

 

Question 167 167 of 569 selected Systems Of 2 Linear Equations In 2 Variables E

A system of two linear equations is graphed in the xy-plane below.

The figure presents the graph of 2 lines in the x y plane. The numbers negative 4 through 16, in increments of 2, are indicated on the x axis. The numbers negative 4 through 20, in increments of 2, are indicated on the y axis. One line begins above the x axis and to the left of the y axis. It moves upward and to the right, crossing the y axis at 6, and passing through the point with coordinates 3 comma 9 and the point with coordinates 12 comma 18. The other line begins below the x axis and to the left of the y axis. It moves upward and to the right, crossing the y axis at 3, and passing through the point with coordinates 3 comma 9 and the point with coordinates 6 comma 15. The two lines intersect at the point with coordinates 3 comma 9.

Which of the following points is the solution to the system of equations?

  1. the point with coordinates 3 comma 9

  2. the point with coordinates 6 comma 15

  3. the point with coordinates 8 comma 10

  4. the point with coordinates 12 comma 18

Show Answer Correct Answer: A

Choice A is correct. The solution to this system of linear equations is the point that lies on both lines graphed, or the point of intersection of the two lines. According to the graphs, the point of intersection occurs when x equals 3 and y equals 9, or at the point with coordinates 3 comma 9.

Choices B and D are incorrect. Each of these points lies on one line, but not on both lines in the xy-plane. Choice C is incorrect. This point doesn’t lie on either of the lines graphed in the xy-plane.

 

Question 168 168 of 569 selected Linear Functions M

f(x)=45x+600

The function f gives the monthly fee f(x), in dollars, a facility charges to keep x crates in storage. What is the monthly fee, in dollars, the facility charges to keep 50 crates in storage?

Show Answer Correct Answer: 2850

The correct answer is 2,850. It’s given that the function f(x)=45x+600 gives the monthly fee, in dollars, a facility charges to keep x crates in storage. Substituting 50 for x in this function yields f(50)=45(50)+600, or f(50)=2,850. Therefore, the monthly fee, in dollars, the facility charges to keep 50 crates in storage is 2,850.

Question 169 169 of 569 selected Linear Functions E

The function f is defined by f(x)=80-6x. What is the value of f(7)?

  1. 13

  2. 38

  3. 74

  4. 81

Show Answer Correct Answer: B

Choice B is correct. It’s given that function f is defined by f(x)=80-6x. The value of f(7) can be found by substituting 7 for x in the given function, which yields f(7)=80-6(7), or f(7)=80-42, which is equivalent to f(7)=38. Therefore, the value of f(7) is 38 .

Choice A is incorrect. This is the value of 80-67, not 80-6(7).

Choice C is incorrect. This is the value of 80-6(1), not 80-6(7).

Choice D is incorrect. This is the value of 80-6+7, not 80-6(7).

Question 170 170 of 569 selected Linear Equations In 1 Variable M

If 46=16+2(x-8), what is the value of 2(x-8)?

  1. 16

  2. 23

  3. 30

  4. 38

Show Answer Correct Answer: C

Choice C is correct. Subtracting 16 from both sides of the given equation yields 30=2(x-8). Therefore, the value of 2(x-8) is 30.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 171 171 of 569 selected Linear Functions H

The functions f and g are defined as f(x)= 1 4 x - 9 and g(x)= 3 4 x + 21 . If the function h is defined as h(x)=f(x)+g(x), what is the x-coordinate of the x-intercept of the graph of y=h(x) in the xy-plane?

Show Answer Correct Answer: -12

The correct answer is -12 . It's given that the functions f and g are defined as f(x)=14x-9 and g(x)=34x+21. If the function h is defined as h(x)=f(x)+g(x), then substituting 14x-9 for f(x) and 34x+21 for g(x) in this function yields h(x)=14x-9+34x+21. This can be rewritten as h(x)=44x+12, or h(x)=x+12. The x-intercept of a graph in the xy-plane is the point on the graph where y = 0 . The equation representing the graph of y=h(x) is y=x+12. Substituting 0 for y in this equation yields 0=x+12. Subtracting 12 from both sides of this equation yields -12 = x , or x = -12 . Therefore, the x-coordinate of the x-intercept of the graph of y=h(x) in the xy-plane is -12 .

Question 172 172 of 569 selected Linear Equations In 1 Variable E

A principal used a total of 25 flags that were either blue or yellow for field day. The principal used 20 blue flags. How many yellow flags were used?

  1. 5

  2. 20

  3. 25

  4. 30

Show Answer Correct Answer: A

Choice A is correct. It's given that a principal used a total of 25 blue flags and yellow flags. It's also given that of the 25 flags used, 20 flags were blue. Subtracting the number of blue flags used from the total number of flags used results in the number of yellow flags used. It follows that the number of yellow flags used is 25-20, or 5

Choice B is incorrect. This is the number of blue flags used.

Choice C is incorrect. This is the total number of flags used.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 173 173 of 569 selected Systems Of 2 Linear Equations In 2 Variables H

During a month, Morgan ran r miles at 5 miles per hour and biked b miles at 10 miles per hour. She ran and biked a total of 200 miles that month, and she biked for twice as many hours as she ran. What is the total number of miles that Morgan biked during the month?

  1. 80

  2. 100

  3. 120

  4. 160

Show Answer Correct Answer: D

Choice D is correct. The number of hours Morgan spent running or biking can be calculated by dividing the distance she traveled during that activity by her speed, in miles per hour, for that activity. So the number of hours she ran can be represented by the expression the fraction r over 5, and the number of hours she biked can be represented by the expression the fraction b over 10. It’s given that she biked for twice as many hours as she ran, so this can be represented by the equation the fraction b over 10, end fraction,  equals, 2 times the fraction r over 5, which can be rewritten as b equals 4 r. It’s also given that she ran r miles and biked b miles, and that she ran and biked a total of 200 miles. This can be represented by the equation r plus b, equals 200 . Substituting 4 r for b in this equation yields r plus 4 r, equals 200 , or 5 r equals 200. Solving for r  yields r equals 40. Determining the number of miles she biked, b, can be found by substituting 40 for r in r plus b, equals 200 , which yields 40 plus b, equals 200. Solving for b yields b equals 160.

Choices A, B, and C are incorrect because they don’t satisfy that Morgan biked for twice as many hours as she ran. In choice A, if she biked 80 miles, then she ran 120 miles, which means she biked for 8 hours and ran for 24 hours. In choice B, if she biked 100 miles, then she ran 100 miles, which means she biked for 10 hours and ran for 20 hours. In choice C, if she biked 120 miles, then she ran for 80 miles, which means she biked for 12 hours and ran for 16 hours.

 

Question 174 174 of 569 selected Systems Of 2 Linear Equations In 2 Variables M

x + 3 = - 2 y + 5

x - 3 = 2 y + 7

The solution to the given system of equations is (x,y). What is the value of 2 x ?

  1. -2

  2. 6

  3. 12

  4. 24

Show Answer Correct Answer: C

Choice C is correct. Adding the second equation in the given system to the first equation in the given system yields (x+3)+(x-3)=(-2y+5)+(2y+7). Adding like terms in this equation yields 2x=12. Thus, the value of 2 x is 12 .

Choice A is incorrect. This is the value of y , not 2 x .

Choice B is incorrect. This is the value of x , not 2 x .

Choice D is incorrect and may result from conceptual or calculation errors.

Question 175 175 of 569 selected Linear Equations In 1 Variable E

8 x = 88

What value of x is the solution to the given equation?

  1. 11

  2. 80

  3. 96

  4. 704

Show Answer Correct Answer: A

Choice A is correct. Dividing both sides of the given equation by 8 yields x = 11 . Therefore, 11 is the solution to the given equation.

Choice B is incorrect. This is the solution to the equation x + 8 = 88 .

Choice C is incorrect. This is the solution to the equation x - 8 = 88 .

Choice D is incorrect. This is the solution to the equation x8=88.

Question 176 176 of 569 selected Systems Of 2 Linear Equations In 2 Variables M

The combined original price for a mirror and a vase is $60. After a 25% discount to the mirror and a 45% discount to the vase are applied, the combined sale price for the two items is $39. Which system of equations gives the original price m , in dollars, of the mirror and the original price v , in dollars, of the vase?

  1. m + v = 60

    0.55 m + 0.75 v = 39

  2. m + v = 60

    0.45 m + 0.25 v = 39

  3. m + v = 60

    0.75 m + 0.55 v = 39

  4. m + v = 60

    0.25 m + 0.45 v = 39

Show Answer Correct Answer: C

Choice C is correct. It’s given that m represents the original price, in dollars, of the mirror, and v represents the original price, in dollars, of the vase. It's also given that the combined original price for the mirror and the vase is $60. This can be represented by the equation m+v=60. After a 25% discount to the mirror is applied, the sale price of the mirror is 75% of its original price. This can be represented by the expression 0.75m. After a 45% discount to the vase is applied, the sale price of the vase is 55% of its original price. This can be represented by the expression 0.55v. It’s given that the combined sale price for the two items is $39. This can be represented by the equation 0.75m+0.55v=39. Therefore, the system of equations consisting of the equations m+v=60 and 0.75m+0.55v=39 gives the original price m , in dollars, of the mirror and the original price v , in dollars, of the vase.

Choice A is incorrect. The second equation in this system of equations represents a 45% discount to the mirror and a 25% discount to the vase.

Choice B is incorrect. The second equation in this system of equations represents a 55% discount to the mirror and a 75% discount to the vase.

Choice D is incorrect. The second equation in this system of equations represents a 75% discount to the mirror and a 55% discount to the vase.

Question 177 177 of 569 selected Linear Functions E

The graph shown models the number of candy bars a certain machine wraps with a label in x seconds.

  • The line slants sharply up from left to right.
  • The line passes through the following points:
    • (0 comma 0)
    • (2 comma 80)
    • (4 comma 160)
    • (6 comma 240)

According to the graph, what is the estimated number of candy bars the machine wraps with a label per second?

  1. 2

  2. 40

  3. 78

  4. 80

Show Answer Correct Answer: B

Choice B is correct. For the graph shown, the x-axis represents time, in seconds, and the y-axis represents the number of candy bars wrapped. The slope of a line in the xy-plane is the change in y for each 1 -unit increase in x . It follows that the slope of the graph shown represents the estimated number of candy bars the machine wraps with a label per second. The slope, m , of a line in the xy-plane can be found using any two points, (x1,y1) and (x2,y2), on the line and the slope formula m=y2-y1x2-x1. The graph shown passes through the points (0,0) and (2,80). Substituting these points for (x1,y1) and (x2,y2), respectively, in the slope formula yields m=80-02-0, which is equivalent to m=802, or m = 40 . Therefore, the estimated number of candy bars the machine wraps with a label per second is 40 .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 178 178 of 569 selected Linear Functions M

h(x)=x+b

For the linear function h , b is a constant and h(0)=45. What is the value of b ?

Show Answer Correct Answer: 45

The correct answer is 45 . It’s given that h(0)=45. Therefore, for the given function h , when x = 0 , h(x)=45. Substituting 0 for x and 45 for h(x) in the given function, h(x)=x+b, yields 45=0+b, or 45=b. Therefore, the value of b is 45 .

Question 179 179 of 569 selected Systems Of 2 Linear Equations In 2 Variables M

4 x plus 5 y, equals 100, 
and, 
5 x plus 4 y, equals 62

If the system of equations above has solution the ordered pair x comma y , what is the value of x plus y ?

  1. 0

  2. 9

  3. 18

  4. 38

Show Answer Correct Answer: C

Choice C is correct. Adding the given equations yields 9x + 9y = 162. Dividing each side of the equation 9x + 9y = 162 by 9 gives x + y = 18.

Choice A is incorrect and may result from incorrectly adding the equations. Choice B is incorrect and may result from conceptual or computational errors. Choice D is incorrect. This value is equivalent to yx.

 

Question 180 180 of 569 selected Systems Of 2 Linear Equations In 2 Variables M

2 a + 8 b = 198

2 a + 4 b = 98

The solution to the given system of equations is (a,b). What is the value of b ?

Show Answer Correct Answer: 25

The correct answer is 25 . Subtracting the second equation from the first equation in the given system of equations yields (2a-2a)+(8b-4b)=198-98, which is equivalent to 0+4b=100, or 4b=100. Dividing each side of this equation by 4 yields b = 25

Question 181 181 of 569 selected Linear Functions E

  • The line slants sharply up from left to right.
  • The line passes through the following points:
    • (0 comma negative 4)
    • (1 comma 0)

The graph of the function f is shown, where y=f(x). What is the y-intercept of the graph?

  1. (0,-1)

  2. (0,-4)

  3. (0,1)

  4. (0,4)

Show Answer Correct Answer: B

Choice B is correct. The y-intercept of a graph is the point where the graph intersects the y-axis. The graph of function f shown intersects the y-axis at the point (0,-4). Therefore, the y-intercept of the graph is (0,-4).

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 182 182 of 569 selected Linear Equations In 2 Variables M
x 1 2 3
y 11 16 21

The table shows three values of x and their corresponding values of y . Which equation represents the linear relationship between x and y ?

  1. y = 5 x + 6

  2. y = 5 x + 11

  3. y = 6 x + 5

  4. y = 6 x + 11

Show Answer Correct Answer: A

Choice A is correct. The linear relationship between x and y can be represented by the equation y = m x + b , where m is the slope of the line in the xy-plane that represents the relationship, and b is the y-coordinate of the y-intercept. The slope can be computed using any two points on the line. The slope of a line between any two points, (x1,y1) and (x2,y2), on the line can be calculated using the slope formula, m=y2-y1x2-x1. In the given table, each value of x and its corresponding value of y can be represented by a point (x,y). In the given table, when the value of x is 1 , the corresponding value of y is 11 and when the value of x is 2 , the corresponding value of y is 16 . Therefore, the points (1,11) and (2,16) are on the line. Substituting (1,11) and (2,16) for (x1,y1) and (x2,y2), respectively, in the slope formula yields m=16-112-1, or m = 5 . Substituting 5 for m in the equation y = m x + b yields y = 5 x + b . Substituting the first value of x in the table, 1 , and its corresponding value of y , 11 , for x and y , respectively, in this equation yields 11=5(1)+b, or 11 = b + 5 . Subtracting 5 from both sides of this equation yields 6 = b . Substituting 6 for b in the equation y = 5 x + b yields y = 5 x + 6 . Therefore, the equation y = 5 x + 6 represents the linear relationship between x and y .

Choice B is incorrect. For this relationship, when the value of x is 1 , the corresponding value of y is 16 , not 11 .

Choice C is incorrect. For this relationship, when the value of x is 2 , the corresponding value of y is 17 , not 16 .

Choice D is incorrect. For this relationship, when the value of x is 1 , the corresponding value of y is 17 , not 11 .

Question 183 183 of 569 selected Linear Equations In 2 Variables M
x y
-6 65
-3 56
3 38
6 29

The table shows four values of x and their corresponding values of y . There is a linear relationship between x and y . Which of the following equations represents this relationship?

  1. 9 x + 3 y = 141

  2. 9 x + 3 y = 3

  3. 3 x + 9 y = 141

  4. 3 x + 9 y = 3

Show Answer Correct Answer: A

Choice A is correct. An equation representing the linear relationship between x and y can be written in slope-intercept form y=mx+b, where m is the slope of the graph of the equation in the xy-plane and (0,b) is the y-intercept. The slope, m, can be calculated using two ordered pairs, (x1,y1) and (x2,y2), and the formula m=y2-y1x2-x1. Substituting the ordered pairs (-6,65) and (6,29) from the table for (x1,y1) and (x2,y2), respectively, in this formula yields m=29-656-(-6), which is equivalent to m=-3612, or m=-3. Substituting -3 for m in the formula y=mx+b yields y=-3x+b. Substituting the point (-6,65) into this equation yields 65=-3(-6)+b, or 65=18+b. Subtracting 18 from both sides of this equation yields 47=b. Substituting 47 for b in the equation y=-3x+b yields y=-3x+47. Adding 3x to both sides of this equation yields 3x+y=47. Multiplying both sides of this equation by 3 yields 9x+3y=141.

Choice B is incorrect. Substituting the point (-6,65) from the table into this equation yields 9(-6)+3(65)=3, or 141=3, which is false.

Choice C is incorrect. Substituting the point (-6,65) from the table into this equation yields 3(-6)+9(65)=141, or 567=141, which is false.

Choice D is incorrect. Substituting the point (-6,65) from the table into this equation yields 3(-6)+9(65)=3, or 567=3, which is false.

Question 184 184 of 569 selected Linear Inequalities In 1 Or 2 Variables H

A number x is at most 2 less than 3 times the value of y . If the value of y is -4 , what is the greatest possible value of x ?

Show Answer Correct Answer: -14

The correct answer is -14 . It's given that a number x is at most 2 less than 3 times the value of y . Therefore, x is less than or equal to 2 less than 3 times the value of y . The expression 3 y represents 3 times the value of y . The expression 3 y - 2 represents 2 less than 3 times the value of y . Therefore, x is less than or equal to 3 y - 2 . This can be shown by the inequality x3y-2. Substituting -4 for y in this inequality yields x3(-4)-2 or, x-14. Therefore, if the value of y is -4 , the greatest possible value of x is -14 .

Question 185 185 of 569 selected Linear Functions M

Caleb used juice to make popsicles. The function f(x)=-5x+30 approximates the volume, in fluid ounces, of juice Caleb had remaining after making x popsicles. Which statement is the best interpretation of the y-intercept of the graph of y=f(x) in the xy-plane in this context?

  1. Caleb used approximately 5 fluid ounces of juice for each popsicle.

  2. Caleb had approximately 5 fluid ounces of juice when he began to make the popsicles.

  3. Caleb had approximately 30 fluid ounces of juice when he began to make the popsicles.

  4. Caleb used approximately 30 fluid ounces of juice for each popsicle.

Show Answer Correct Answer: C

Choice C is correct. An equation that defines a linear function f can be written in the form f(x)=mx+b, where m represents the slope and b represents the y-intercept, (0,b), of the line of y=f(x) in the xy-plane. The function f(x)=-5x+30 is linear. Therefore, the graph of the given function y=f(x) in the xy-plane has a y-intercept of (0,30). It’s given that f(x) gives the approximate volume, in fluid ounces, of juice Caleb had remaining after making x popsicles. It follows that the y-intercept of (0,30) means that Caleb had approximately 30 fluid ounces of juice remaining after making 0 popsicles. In other words, Caleb had approximately 30 fluid ounces of juice when he began to make the popsicles.

Choice A is incorrect. This is an interpretation of the slope, rather than the y-intercept, of the graph of y=f(x) in the xy-plane.

Choice B is incorrect and may result from conceptual errors.

Choice D is incorrect and may result from conceptual errors.

Question 186 186 of 569 selected Linear Equations In 2 Variables M

At a state fair, attendees can win tokens that are worth a different number of points depending on the shape. One attendee won S square tokens and C circle tokens worth a total of 1,120 points. The equation 80S+90C=1,120 represents this situation. How many more points is a circle token worth than a square token?

  1. 950

  2. 90

  3. 80

  4. 10

Show Answer Correct Answer: D

Choice D is correct. It’s given that the equation 80S+90C=1,120 represents this situation, where S is the number of square tokens won, C is the number of circle tokens won, and 1,120 is the total number of points the tokens are worth. It follows that 80 S represents the total number of points the square tokens are worth. Therefore, each square token is worth 80 points. It also follows that 90 C represents the total number of points the circle tokens are worth. Therefore, each circle token is worth 90 points. Since a circle token is worth 90 points and a square token is worth 80 points, then a circle token is worth 90-80, or 10 , more points than a square token.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect. This is the number of points a circle token is worth.

Choice C is incorrect. This is the number of points a square token is worth.

Question 187 187 of 569 selected Linear Functions H

According to data provided by the US Department of Energy, the average price per gallon of regular gasoline in the United States from September 1, 2014, to December 1, 2014, is modeled by the function F defined below, where F of x is the average price per gallon x months after September 1.

The constant 2.74 in this function estimates which of the following?

  1. The average monthly decrease in the price per gallon

  2. The difference in the average price per gallon from September 1, 2014, to December 1, 2014

  3. The average price per gallon on September 1, 2014

  4. The average price per gallon on December 1, 2014

Show Answer Correct Answer: D

Choice D is correct. Since 2.74 is a constant term, it represents an actual price of gas rather than a measure of change in gas price. To determine what gas price it represents, find x such that F(x) = 2.74, or 2.74 = 2.74 – 0.19(x – 3). Subtracting 2.74 from both sides gives 0 = –0.19(x – 3). Dividing both sides by –0.19 results in 0 = x – 3, or x = 3. Therefore, the average price of gas is $2.74 per gallon 3 months after September 1, 2014, which is December 1, 2014.

Choice A is incorrect. Since 2.74 is a constant, not a multiple of x, it cannot represent a rate of change in price. Choice B is incorrect. The difference in the average price from September 1, 2014, to December 1, 2014, is F(3) – F(0) = 2.74 – 0.19(3 – 3) – (2.74 – 0.19(0 – 3)) = 2.74 – (2.74 + 0.57) = –0.57, which is not 2.74. Choice C is incorrect. The average price per gallon on September 1, 2014, is F(0) = 2.74 – 0.19(0 – 3) = 2.74 + 0.57 = 3.31, which is not 2.74.

Question 188 188 of 569 selected Systems Of 2 Linear Equations In 2 Variables M

In the xy-plane, the graph of y equals, x plus 3 intersects the graph of y equals, 2 x minus 6 at the point with coordinates a, comma b. What is the value of a ?

  1. 3

  2. 6

  3. 9

  4. 12

Show Answer Correct Answer: C

Choice C is correct. Since the graph of y equals, x plus 3 intersects the graph of y equals, 2 x minus 6 at the point with coordinates a, comma b, the ordered pair a, comma b is the solution to the system of linear equations consisting of y equals, x plus 3 and y equals, 2 x minus 6, and the value of a is the value of x in the solution of this system. Since both x plus 3 and 2 x minus 6 are equal to y, it follows that x plus 3, equals, 2 x minus 6. Subtracting x from and adding 6 to both sides of the equation yields 9 equals x. Therefore, the value of a is 9.

Choices A and B are incorrect and may result from a calculation or conceptual error in solving the system of equations consisting of y equals, x plus 3 and y equals, 2 x minus 6. Choice D is incorrect. This is the value of b, not a.

Question 189 189 of 569 selected Linear Functions M

In the xy-plane, the graph of the linear function f contains the points (0,2) and (8,34). Which equation defines f , where y=f(x)?

  1. f(x)=2x+42

  2. f(x)=32x+36

  3. f(x)=4x+2

  4. f(x)=8x+2

Show Answer Correct Answer: C

Choice C is correct. In the xy-plane, the graph of a linear function can be written in the form f(x)=mx+b, where m represents the slope and (0,b) represents the y-intercept of the graph of y=f(x). It’s given that the graph of the linear function f , where y=f(x), in the xy-plane contains the point (0,2). Thus, b = 2 . The slope of the graph of a line containing any two points (x1,y1) and (x2,y2) can be found using the slope formula, m=y2-y1x2-x1. Since it’s given that the graph of the linear function f contains the points (0,2) and (8,34), it follows that the slope of the graph of the line containing these points is m=34-28-0, or  m = 4 . Substituting 4 for m and 2 for b in f(x)=mx+b yields f(x)=4x+2.

Choice A is incorrect. This function represents a graph with a slope of 2 and a y-intercept of (0,42).

Choice B is incorrect. This function represents a graph with a slope of 32 and a y-intercept of (0,36).

Choice D is incorrect. This function represents a graph with a slope of 8 and a y-intercept of (0,2).

Question 190 190 of 569 selected Linear Functions M
Number of cars Maximum number of passengers and crew
3 174
5 284
10 559

The table shows the linear relationship between the number of cars, c , on a commuter train and the maximum number of passengers and crew, p , that the train can carry. Which equation represents the linear relationship between c and p ?

  1. 55c-p=-9

  2. 55c-p=9

  3. 55p-c=-9

  4. 55p-c=9

Show Answer Correct Answer: A

Choice A is correct. It's given that there is a linear relationship between the number of cars, c , on a commuter train and the maximum number of passengers and crew, p , that the train can carry. It follows that this relationship can be represented by an equation of the form p=mc+b, where m is the rate of change of p in this relationship and b is a constant. The rate of change of p in this relationship can be calculated by dividing the difference in any two values of p by the difference in the corresponding values of c . Using two pairs of values given in the table, the rate of change of p in this relationship is 284-1745-3, or 55 . Substituting 55 for m in the equation p=mc+b yields p=55c+b. The value of b can be found by substituting any value of c and its corresponding value of p for c and p , respectively, in this equation. Substituting 10 for c and 559 for p yields 559=55(10)+b, or 559=550+b. Subtracting 550 from both sides of this equation yields 9=b. Substituting 9 for b in the equation p=55c+b yields p=55c+9. Subtracting 9 from both sides of this equation yields p-9=55c. Subtracting p from both sides of this equation yields -9=55c-p, or 55c-p=-9.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 191 191 of 569 selected Linear Equations In 2 Variables E

The y-intercept of the graph of 12 x + 2 y = 18 in the xy-plane is (0,y). What is the value of y ?

Show Answer Correct Answer: 9

The correct answer is 9 . It's given that the y-intercept of the graph of 12 x + 2 y = 18 in the xy-plane is (0,y). Substituting 0 for x in the equation 12 x + 2 y = 18 yields 12(0)+2y=18, or 2 y = 18 . Dividing both sides of this equation by 2 yields y = 9 . Therefore, the value of y is 9 .

Question 192 192 of 569 selected Linear Equations In 2 Variables E

  • The line slants sharply up from left to right.
  • The line passes through the following points:
    • (0 comma 5)
    • (1 comma 9)

Line j is shown in the xy-plane. Line k (not shown) is parallel to line j . What is the slope of line k ?

Show Answer Correct Answer: 4

The correct answer is 4 . It's given that line k is parallel to line j . It follows that the slope of line k is equal to the slope of line j . Given two points on a line in the xy-plane, (x1,y1) and (x2,y2), the slope of the line can be calculated as y2-y1x2-x1. In the xy-plane shown, the points (0,5) and (1,9) are on line j . It follows that the slope of line j is 9-51-0, or 4 . Since the slope of line j is equal to the slope of line k , the slope of line k is also 4 .

Question 193 193 of 569 selected Linear Equations In 2 Variables H

In the xy-plane, line l passes through the point (0,0) and is parallel to the line represented by the equation y = 8 x + 2 . If line l also passes through the point (3,d), what is the value of d ?

Show Answer Correct Answer: 24

The correct answer is 24 . A line in the xy-plane can be defined by the equation y = m x + b , where m is the slope of the line and b is the y-coordinate of the y-intercept of the line. It's given that line l passes through the point (0,0). Therefore, the y-coordinate of the y-intercept of line l is 0 . It's given that line l is parallel to the line represented by the equation y=8x+2. Since parallel lines have the same slope, it follows that the slope of line l is 8 . Therefore, line l can be defined by an equation in the form y = m x + b , where m = 8 and b = 0 . Substituting 8 for m and 0 for b in y = m x + b yields the equation y=8x+0, or y = 8 x . If line l passes through the point (3,d), then when x = 3 , y = d for the equation y = 8 x . Substituting 3 for x and d for y in the equation y = 8 x yields d=8(3), or d = 24 .

Question 194 194 of 569 selected Linear Functions E

f(x)=x+811

The function f is defined by the given equation. What is the value of f(x) when x = 3 11 ?

Show Answer Correct Answer: 1

The correct answer is 1 . It’s given that the function f is defined by f(x)=x+811. Substituting 311for x in the given function yields f(311)=311+811, which gives f(311)=1111, or f(311)=1. Therefore, when x=311, the value of f(x) is 1 .

Question 195 195 of 569 selected Linear Functions M

Brian saves 2 5 of the $215 he earns each week from his job. If Brian continues to save at this rate, how much money, in dollars, will Brian save in 9 weeks?

Show Answer Correct Answer: 774

The correct answer is 774 . It's given that Brian saves 2 5 of the $215 he earns each week from his job. Therefore, Brian saves (25)($215), or $86, per week. If Brian continues to save at this rate of $86 per week for 9 weeks, then he will save a total of (9)(86), or 774 , dollars.

Question 196 196 of 569 selected Linear Equations In 2 Variables M

What is the slope of the graph of y=13(29x+10)+5x in the xy-plane?

Show Answer Correct Answer: 14.66, 14.67, 44/3

The correct answer is 44 3 . A linear equation can be written in the form y=mx+b, where m is the slope of the graph of the equation in the xy-plane and (0,b) is the y-intercept. Distributing the 1 3 in the equation y=13(29x+10)+5x yields y=293x+103+5x. Combining like terms on the right-hand side of this equation yields y=443x+103. This equation is in the form y=mx+b, where m = 44 3 and b = 10 3 . Therefore, the slope of the graph of the given equation in the xy-plane is 44 3 . Note that 44/3, 14.66, and 14.67 are examples of ways to enter a correct answer.

Question 197 197 of 569 selected Linear Functions H

One gallon of stain will cover 170 square feet of a surface. A yard has a total fence area of w  square feet. Which equation represents the total amount of stain S , in gallons, needed to stain the fence in this yard twice?

  1. S = w 170

  2. S = 170 w

  3. S = 340 w

  4. S = w 85

Show Answer Correct Answer: D

Choice D is correct. It's given that w represents the total fence area, in square feet. Since the fence will be stained twice, the amount of stain, in gallons, will need to cover 2 w square feet. It’s also given that one gallon of stain will cover 170 square feet. Dividing the total area, in square feet, of the surface to be stained by the number of square feet covered by one gallon of stain gives the number of gallons of stain that will be needed. Dividing 2 w by 170 yields  2w170, or w85. Therefore, the equation that represents the total amount of stain S , in gallons, needed to stain the fence of the yard twice is S=w85.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Question 198 198 of 569 selected Linear Functions M

For the linear function h , the graph of y=h(x) in the xy-plane passes through the points (7,21) and (9,25). Which equation defines h ?

  1. h(x)=12x-72

  2. h(x)=2x+7

  3. h(x)=7x+21

  4. h(x)=9x+25

Show Answer Correct Answer: B

Choice B is correct. It’s given that the graph of the linear function h , where y=h(x), passes through the points (7,21) and (9,25) in the xy-plane. An equation defining h can be written in the form y=mx+b, where y=h(x), m represents the slope of the graph in the xy-plane, and b represents the y-coordinate of the y-intercept of the graph. The slope can be found using any two points, (x1,y1) and (x2,y2), and the formula m=(y2-y1)(x2-x1). Substituting (7,21) and (9,25) for (x1,y1) and (x2,y2), respectively, in the slope formula yields m=25-219-7, which is equivalent to m=42, or m = 2 . Substituting 2 for m and (7,21) for (x,y) in the equation y=mx+b yields 21=(2)(7)+b, or 21=14+b. Subtracting 14 from each side of this equation yields 7=b. Substituting 2 for m and 7 for b in the equation y=mx+b yields y=2x+7. Since y=h(x), it follows that the equation that defines h is h(x)=2x+7.

Choice A is incorrect. For this function, the graph of y=h(x) in the xy-plane would pass through (7,0), not (7,21), and (9,1), not (9,25).

Choice C is incorrect. For this function, the graph of y=h(x) in the xy-plane would pass through (7,70), not (7,21), and (9,84), not (9,25).

Choice D is incorrect. For this function, the graph of y=h(x) in the xy-plane would pass through (7,88), not (7,21), and (9,106), not (9,25).

Question 199 199 of 569 selected Linear Functions E

f(x)=8x+4

The function f gives the estimated height, in feet, of a willow tree x years after its height was first measured. Which statement is the best interpretation of 4 in this context?

  1. The tree will be measured each year for 4 years.

  2. The tree is estimated to grow to a maximum height of 4 feet.

  3. The estimated height of the tree increased by 4 feet each year. 

  4. The estimated height of the tree was 4 feet when it was first measured.

Show Answer Correct Answer: D

Choice D is correct. It's given that the function f(x)=8x+4 gives the estimated height, in feet, of a willow tree x years after its height was first measured. For a function defined by an equation of the form f(x)=mx+b, where m and b are constants, b represents the value of f(x) when x = 0 . It follows that in the given function, 4 represents the value of f(x) when x = 0 . Therefore, the best interpretation of 4 in this context is that the estimated height of the tree was 4 feet when it was first measured.

Choice A is incorrect and may result from conceptual errors.

Choice B is incorrect and may result from conceptual errors.

Choice C is incorrect and may result from conceptual errors.

Question 200 200 of 569 selected Linear Inequalities In 1 Or 2 Variables E

Tom scored 85, 78, and 98 on his first three exams in history class. Solving which inequality gives the score, G, on Tom’s fourth exam that will result in a mean score on all four exams of at least 90 ?

  1. 90 minus, open parenthesis, 85 plus 78 plus 98, close parenthesis, is less than or equal to 4 G

  2. 4 G, plus 85, plus 78, plus 98, is greater than or equal to 360

  3. the fraction with numerator, open parenthesis, G plus 85 plus 78 plus 98, close parenthesis, and denominator 4, end fraction, is greater than or equal to 90

  4. the fraction with numerator, open parenthesis, 85 plus 78 plus 98, close parenthesis, and denominator 4, end fraction, is greater than or equal to 90 minus 4 G

Show Answer Correct Answer: C

Choice C is correct. The mean of the four scores (G, 85, 78, and 98) can be expressed as  the fraction with numerator G, plus 85, plus 78, plus 98, and denominator 4. The inequality that expresses the condition that the mean score is at least 90 can therefore be written as the fraction with numerator G, plus 85, plus 78, plus 98, and denominator 4, is greater than or equal to 90.

Choice A is incorrect. The sum of the scores (G, 85, 78, and 98) isn’t divided by 4 to express the mean. Choice B is incorrect and may be the result of an algebraic error when multiplying both sides of the inequality by 4. Choice D is incorrect because it doesn’t include G in the mean with the other three scores.

 

Question 201 201 of 569 selected Systems Of 2 Linear Equations In 2 Variables H

 

In the system of equations below, a and c are constants.

Equation 1: one half x, plus one third y, equals one sixth. Equation 2: a, x, plus y equals c

If the system of equations has an infinite number of solutions x comma y , what is the value of a ?

 

  1. negative one half

  2. 0

  3. one half

  4. three halves

Show Answer Correct Answer: D

Choice D is correct. A system of two linear equations has infinitely many solutions if one equation is equivalent to the other. This means that when the two equations are written in the same form, each coefficient or constant in one equation is equal to the corresponding coefficient or constant in the other equation multiplied by the same number. The equations in the given system of equations are written in the same form, with x and y on the left-hand side and a constant on the right-hand side of the equation. The coefficient of y in the second equation is equal to the coefficient of y in the first equation multiplied by 3. Therefore, a, the coefficient of x in the second equation, must be equal to 3 times the coefficient of x in the first equation: a, equals, one half times 3, or a, equals three halves.

Choices A, B, and C are incorrect. When a, equals negative one half, a, equals 0, or a, equals one half, the given system of equations has one solution.

 

Question 202 202 of 569 selected Linear Functions E

Sean rents a tent at a cost of $11 per day plus a onetime insurance fee of $10. Which equation represents the total cost c , in dollars, to rent the tent with insurance for d days?

  1. c=11(d+10)

  2. c=10(d+11)

  3. c = 11 d + 10

  4. c = 10 d + 11

Show Answer Correct Answer: C

Choice C is correct. It’s given that the cost of renting a tent is $11 per day for d days. Multiplying the rental cost by the number of days yields $11d, which represents the cost of renting the tent for d days before the insurance is added. Adding the onetime insurance fee of $10 to the rental cost of $11d gives the total cost c , in dollars, which can be represented by the equation c=11d+10.

Choice A is incorrect. This equation represents the total cost to rent the tent if the insurance fee was charged every day.

Choice B is incorrect. This equation represents the total cost to rent the tent if the daily fee was $(d+11) for 10 days.

Choice D is incorrect. This equation represents the total cost to rent the tent if the daily fee was $10 and the onetime fee was $11.

Question 203 203 of 569 selected Linear Equations In 2 Variables H

  • The line slants gradually down from left to right.
  • The line passes through the following points:
    • (negative 8 comma 4)
    • (0 comma 2)
    • (8 comma 0)

The graph of y=f(x)+14 is shown. Which equation defines function f ?

  1. f(x)= - 1 4 x - 12

  2. f(x)= - 1 4 x + 16

  3. f(x)= - 1 4 x + 2

  4. f(x)= - 1 4 x - 14

Show Answer Correct Answer: A

Choice A is correct. An equation for the graph shown can be written in slope-intercept form y = m x + b , where m is the slope of the graph and its y-intercept is (0,b). Since the y-intercept of the graph shown is (0,2), the value of b is 2 . Since the graph also passes through the point (4,1), the slope can be calculated as 1-24-0, or - 1 4 . Therefore, the value of m is - 1 4 . Substituting - 1 4 for m and 2 for b in the equation y = m x + b yields y=-14x+2. It’s given that an equation for the graph shown is y=f(x)+14. Substituting f(x)+14 for y in the equation y=-14x+2  yields f(x)+14=-14x+2. Subtracting 14 from both sides of this equation yields f(x)=-14x-12.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 204 204 of 569 selected Linear Equations In 2 Variables H

Keenan made 32 cups of vegetable broth. Keenan then filled x small jars and y large jars with all the vegetable broth he made. The equation 3x+5y=32 represents this situation. Which is the best interpretation of 5 y in this context?

  1. The number of large jars Keenan filled

  2. The number of small jars Keenan filled

  3. The total number of cups of vegetable broth in the large jars

  4. The total number of cups of vegetable broth in the small jars

Show Answer Correct Answer: C

Choice C is correct. It’s given that the equation 3 x + 5 y = 32 represents the situation where Keenan filled x small jars and y large jars with all the vegetable broth he made, which was 32 cups. Therefore, 3 x represents the total number of cups of vegetable broth in the small jars and 5 y represents the total number of cups of vegetable broth in the large jars. 

Choice A is incorrect. The number of large jars Keenan filled is represented by y , not 5 y .

Choice B is incorrect. The number of small jars Keenan filled is represented by x , not 5 y .

Choice D is incorrect. The total number of cups of vegetable broth in the small jars is represented by 3 x , not 5 y

Question 205 205 of 569 selected Linear Inequalities In 1 Or 2 Variables E

During spring migration, a dragonfly traveled a minimum of 1,510 miles and a maximum of 4,130 miles between stopover locations. Which inequality represents this situation, where d is a possible distance, in miles, this dragonfly traveled between stopover locations during spring migration?

  1. d1,510

  2. 1,510d4,130

  3. d4,130

  4. 4,130d5,640

Show Answer Correct Answer: B

Choice B is correct. It's given that during spring migration, a dragonfly traveled a minimum of 1,510 miles and a maximum of 4,130 miles between stopover locations. It's also given that d represents a possible distance, in miles, this dragonfly traveled between stopover locations. It follows that the inequality 1,510d4,130 represents this situation.

Choice A is incorrect. This inequality represents a situation in which a dragonfly traveled a maximum of 1,510 miles between stopover locations.

Choice C is incorrect. This inequality represents a situation in which a dragonfly traveled a minimum of 4,130 miles between stopover locations.

Choice D is incorrect. This inequality represents a situation in which a dragonfly traveled a minimum of 4,310 miles and a maximum of 5,640 miles between stopover locations.

Question 206 206 of 569 selected Linear Functions M

The boiling point of water at sea level is 212 degrees Fahrenheit degrees Fahrenheit. For every 550 feet above sea level, the boiling point of water is lowered by about 1 degree Fahrenheit. Which of the following equations can be used to find the boiling point B of water, in degrees Fahrenheit, x feet above sea level?

  1. B equals, 550 plus, the fraction x over 212

  2. B equals, 550 minus, the fraction x over 212

  3. B equals, 212 plus, the fraction x over 550

  4. B equals, 212 minus, the fraction x over 550

Show Answer Correct Answer: D

Choice D is correct. It’s given that the boiling point of water at sea level is 212°F and that for every 550 feet above sea level, the boiling point of water is lowered by about 1°F. Therefore, the change in the boiling point of water x feet above sea level is represented by the expression the negative of the fraction x over 550. Adding this expression to the boiling point of water at sea level gives the equation for the boiling point B of water, in °F, x feet above sea level: B equals, the negative of the fraction x over 550, end fraction, plus 212, or B equals, 212 minus, the fraction x over 550.

Choices A and B are incorrect and may result from using the boiling point of water at sea level as the rate of change and the rate of change as the initial boiling point of water at sea level. Choice C is incorrect and may result from representing the change in the boiling point of water as an increase rather than a decrease.

 

Question 207 207 of 569 selected Linear Inequalities In 1 Or 2 Variables M

y<4x+4

Which point (x , y ) is a solution to the given inequality in the x y -plane?

  1. (-4 , 0 )

  2. (0 , 5 )

  3. (2 , 1 )

  4. (2 , -1 )

Show Answer Correct Answer: A

Choice D is correct. For a point (x,y) to be a solution to the given inequality in the xy-plane, the value of the point’s y-coordinate must be less than the value of -4x+4, where x is the value of the x-coordinate of the point. This is true of the point (-4,0) because 0<-4(-4)+4, or 0<20. Therefore, the point (-4,0) is a solution to the given inequality.


Choices A, B, and C are incorrect. None of these points are a solution to the given inequality because each point’s y-coordinate is greater than the value of -4x+4 for the point’s x-coordinate.

Question 208 208 of 569 selected Linear Functions E
The figure presents a 2-column table with 3 rows of data. The heading for the first column is “x.” The heading for the second column is “f of x.” The three rows of data are as follows.

Row 1. x, 1; f of x, 5.
Row 2. x, 3; f of x, 13.
Row 3. x, 5; f of x, 21.

Some values of the linear function f are shown in the table above. Which of the following defines f ?

  1. f of x equals, 2 x plus 3

  2. f of x equals, 3 x plus 2

  3. f of x equals, 4 x plus 1

  4. f of x equals, 5 x

Show Answer Correct Answer: C

Choice C is correct. Because f is a linear function of x, the equation f of x equals, m x plus b, where m and b are constants, can be used to define the relationship between x and f (x). In this equation, m represents the increase in the value of f (x) for every increase in the value of x by 1. From the table, it can be determined that the value of f (x) increases by 8 for every increase in the value of x by 2. In other words, for the function f the value of m is eight halves, or 4. The value of b can be found by substituting the values of x and f (x) from any row of the table and the value of m into the equation f of x equals, m x plus b and solving for b. For example, using x equals 1, f of x equals 5, and m equals 4 yields 5 equals, 4 times 1, plus b. Solving for b yields b equals 1. Therefore, the equation defining the function f can be written in the form f of x equals, 4 x plus 1.

Choices A, B, and D are incorrect. Any equation defining the linear function f must give values of f (x) for corresponding values of x, as shown in each row of the table. According to the table, if x equals 3, f of x equals 13. However, substituting x equals 3 into the equation given in choice A gives f of 3 equals, 2 times 3, plus 3, or f of 3 equals 9, not 13. Similarly, substituting x equals 3 into the equation given in choice B gives f of 3 equals, 3 times 3, plus 2, or f of 3 equals 11, not 13.

Lastly, substituting x equals 3 into the equation given in choice D gives f of 3 equals, 5 times 3, or f of 3 equals 15, not 13. Therefore, the equations in choices A, B, and D cannot define f.

 

Question 209 209 of 569 selected Linear Functions E

f(x)=14+4x

The function f represents the total cost, in dollars, of attending an arcade when x games are played. How many games can be played for a total cost of $58?

Show Answer Correct Answer: 11

The correct answer is 11 . It’s given that the function f(x)=14+4x represents the total cost, in dollars, of attending an arcade when x games are played. Substituting 58 for f(x) in the given equation yields 58=14+4x. Subtracting 14 from each side of this equation yields 44=4x. Dividing each side of this equation by 4 yields 11 = x . Therefore, 11 games can be played for a total cost of $58.

Question 210 210 of 569 selected Linear Equations In 2 Variables E

  • The line slants sharply down from left to right.
  • The line passes through the following points:
    • (0 comma 18)
    • (4 comma 10)
    • (7 comma 4)
    • (9 comma 0)

The graph shows the possible combinations of the number of pounds of tangerines and lemons that could be purchased for $18 at a certain store. If Melvin purchased lemons and 4 pounds of tangerines for a total of $18, how many pounds of lemons did he purchase?

  1. 7

  2. 10

  3. 14

  4. 16

Show Answer Correct Answer: B

Choice B is correct. It's given that the graph shows the possible combinations of the number of pounds of tangerines, x, and the number of pounds of lemons, y, that could be purchased for $18 at a certain store. If Melvin purchased lemons and 4 pounds of tangerines for a total of $18, the number of pounds of lemons he purchased is represented by the y-coordinate of the point on the graph where x=4. For the graph shown, when x=4, y=10. Therefore, if Melvin purchased lemons and 4 pounds of tangerines for a total of $18, then he purchased 10 pounds of lemons.

Choice A is incorrect. This is the number of pounds of tangerines Melvin purchased if he purchased tangerines and 4 pounds of lemons for a total of $18.

Choice C is incorrect. This is the number of pounds of lemons Melvin purchased if he purchased lemons and 2 pounds of tangerines for a total of $18.

Choice D is incorrect. This is the number of pounds of lemons Melvin purchased if he purchased lemons and 1 pound of tangerines for a total of $18.

Question 211 211 of 569 selected Linear Equations In 1 Variable H

A certain product costs a company $65 to make. The product is sold by a salesperson who earns a commission that is equal to 20% of the sales price of the product. The profit the company makes for each unit is equal to the sales price minus the combined cost of making the product and the commission. If the sales price of the product is $100, which of the following equations gives the number of units, u, of the product the company sold to make a profit of $6,840 ?

  1. 0 point 8 times 100, minus 65 u, equals 6,840

Show Answer Correct Answer: A

Choice A is correct. The sales price of one unit of the product is given as $100. Because the salesperson is awarded a commission equal to 20% of the sales price, the expression 100(1 – 0.2) gives the sales price of one unit after the commission is deducted. It is also given that the profit is equal to the sales price minus the combined cost of making the product, or $65, and the commission: 100(1 – 0.2) – 65. Multiplying this expression by u gives the profit of u units: (100(1 – 0.2) – 65)u. Finally, it is given that the profit for u units is $6,840; therefore (100(1 – 0.2) – 65)u = $6,840.

Choice B is incorrect. In this equation, cost is subtracted before commission and the equation gives the commission, not what the company retains after commission. Choice C is incorrect because the number of units is multiplied only by the cost but not by the sale price. Choice D is incorrect because the value 0.2 shows the commission, not what the company retains after commission.

 

Question 212 212 of 569 selected Linear Functions E

A bus is traveling at a constant speed along a straight portion of road. The equation d = 30 t gives the distance d , in feet from a road marker, that the bus will be t seconds after passing the marker. How many feet from the marker will the bus be 2 seconds after passing the marker?

  1. 30

  2. 32

  3. 60

  4. 90

Show Answer Correct Answer: C

Choice C is correct. It’s given that t represents the number of seconds after the bus passes the marker. Substituting 2 for t in the given equation d = 30 t yields d=30(2), or d = 60 . Therefore, the bus will be 60 feet from the marker 2 seconds after passing it.

Choice A is incorrect. This is the distance, in feet, the bus will be from the marker 1 second, not 2 seconds, after passing it.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect. This is the distance, in feet, the bus will be from the marker 3 seconds, not 2 seconds, after passing it.

Question 213 213 of 569 selected Systems Of 2 Linear Equations In 2 Variables H

y = 4 x + 1
4 y = 15 x - 8

The solution to the given system of equations is (x,y). What is the value of x-y?

Show Answer Correct Answer: 35

The correct answer is 35 . The first equation in the given system of equations defines y as 4x+1. Substituting 4x+1 for y in the second equation in the given system of equations yields 4(4x+1)=15x-8. Applying the distributive property on the left-hand side of this equation yields 16x+4=15x-8. Subtracting 15 x from each side of this equation yields x+4=-8. Subtracting 4 from each side of this equation yields x=-12. Substituting -12 for x in the first equation of the given system of equations yields y=4(-12)+1, or y=-47. Substituting -12 for x and -47 for y into the expression x-y yields -12-(-47), or 35 .

Question 214 214 of 569 selected Linear Functions E

The cost y , in dollars, for a manufacturer to make x rings is represented by the line shown.

  • The line slants gradually up from left to right.
  • The line passes through the following points:
    • (0 comma 100)
    • (60 comma 175)
    • (100 comma 225)

What is the cost, in dollars, for the manufacturer to make 60 rings?

  1. 100

  2. 125

  3. 175

  4. 225

Show Answer Correct Answer: C

Choice C is correct. The line shown represents the cost y , in dollars, for a manufacturer to make x rings. For the line shown, the x-axis represents the number of rings made by the manufacturer and the y-axis represents the cost, in dollars. Therefore, the cost, in dollars, for the manufacturer to make 60 rings is represented by the y-coordinate of the point on the line that has an x-coordinate of 60 . The point on the line with an x-coordinate of 60 has a y-coordinate of 175 . Therefore, the cost, in dollars, for the manufacturer to make 60 rings is 175 .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect. This is the cost, in dollars, for the manufacturer to make 20 rings.

Choice D is incorrect. This is the cost, in dollars, for the manufacturer to make 100 rings.

Question 215 215 of 569 selected Linear Equations In 1 Variable E

3 more than 8 times a number x is equal to 83. Which equation represents this situation?

  1. (3)(8)x=83

  2. 8x=83+3

  3. 3x+8=83

  4. 8x+3=83

Show Answer Correct Answer: D

Choice D is correct. The given phrase “8 times a number x ” can be represented by the expression 8 x . The given phrase “3 more than” indicates an increase of 3 to a quantity. Therefore “3 more than 8 times a number x ” can be represented by the expression 8 x + 3 . Since it’s given that 3 more than 8 times a number x is equal to 83 , it follows that 8 x + 3 is equal to 83 , or 8 x + 3 = 83 . Therefore, the equation that represents this situation is 8 x + 3 = 83 .

Choice A is incorrect. This equation represents 3 times the quantity 8 times a number x is equal to 83

Choice B is incorrect. This equation represents 8 times a number x is equal to 3 more than 83 .

Choice C is incorrect. This equation represents 8 more than 3 times a number x is equal to 83 .

Question 216 216 of 569 selected Systems Of 2 Linear Equations In 2 Variables H

Equation 1: 7 x minus 5 y, equals 4. Equation 2: 4 x minus 8 y, equals 9.

If the ordered pair x comma y is the solution to the system of equations above, what is the value of 3 x plus 3 y ?

  1. negative 13

  2. negative 5

  3. 5

  4. 13

Show Answer Correct Answer: B

Choice B is correct. Subtracting the second equation, 4 x minus 8 y, equals 9, from the first equation, 7 x minus 5 y, equals 4, results in open parenthesis, 7 x minus 5 y, close parenthesis, minus, open parenthesis, 4 x minus 8 y, close parenthesis, equals, 4 minus 9, or 7 x minus 5 y, minus 4 x, plus 8 y, equals 5. Combining like terms on the left-hand side of this equation yields 3 x plus 3 y, equals negative 5.

Choice A is incorrect and may result from miscalculating 4 minus 9 as negative 13. Choice C is incorrect and may result from miscalculating 4 minus 9 as 5. Choice D is incorrect and may result from adding 9 to 4 instead of subtracting 9 from 4.

 

Question 217 217 of 569 selected Linear Equations In 2 Variables M

  • The line slants gradually down from left to right.
  • The line passes through the following points:
    • (negative 8 comma 0)
    • (0 comma negative 8) 

What is an equation of the graph shown?

  1. y = - 2 x - 8

  2. y = x - 8

  3. y = - x - 8

  4. y = 2 x - 8

Show Answer Correct Answer: C

Choice C is correct. An equation of a line can be written in the form y = m x + b , where m is the slope of the line and (0,b) is the y-intercept of the line. The line shown passes through the point (0,-8), so b=-8. The line shown also passes through the point (-8,0). The slope, m , of a line passing through two points (x1,y1) and (x2,y2) can be calculated using the equation m=y2-y1x2-x1. For the points (0,-8) and (-8,0), this gives m=(-8)-00-(-8), or m=-1. Substituting -8 for b and -1 for m in y = m x + b yields y=(-1)x+(-8), or y=-x-8. Therefore, an equation of the graph shown is y=-x-8.

Choice A is incorrect. This is an equation of a line with a slope of -2 , not -1 .

Choice B is incorrect. This is an equation of a line with a slope of 1 , not -1 .

Choice D is incorrect. This is an equation of a line with a slope of 2 , not -1 .

Question 218 218 of 569 selected Linear Functions E

The function f is defined by f(x)=3x-8. What is the value of f(7)?

  1. 29

  2. 13

  3. -5

  4. -29

Show Answer Correct Answer: B

Choice B is correct. It’s given that the function f is defined by f(x)=3x-8. The value of f(7) is the value of f(x) when x = 7 . Substituting 7 for x in the given equation yields f(7)=3(7)-8, which is equivalent to f(7)=21-8, or f(7)=13.

Choice A is incorrect. This is the value of f(7) when f(x)=3x+8, rather than f(x)=3x-8.

Choice C is incorrect. This is the value of f(1), rather than f(7).

Choice D is incorrect. This is the value of f(-7), rather than f(7).

Question 219 219 of 569 selected Linear Equations In 2 Variables E

A city’s total expense budget for one year was x million dollars. The city budgeted y million dollars for departmental expenses and 201 million dollars for all other expenses. Which of the following represents the relationship between x and y in this context?

  1. x plus y, equals 201

  2. x minus y, equals 201

  3. 2 x minus y, equals 201

  4. y minus x, equals 201

Show Answer Correct Answer: B

Choice B is correct. Of the city’s total expense budget for one year, the city budgeted y million dollars for departmental expenses and 201 million dollars for all other expenses. This means that the expression y plus 201 represents the total expense budget, in millions of dollars, for one year. It’s given that the total expense budget for one year is x million dollars. It follows then that the expression y plus 201 is equivalent to x, or y plus 201, equals x. Subtracting y from both sides of this equation yields 201 equals, x minus y. By the symmetric property of equality, this is the same as x minus y, equals 201.

Choices A and C are incorrect. Because it’s given that the total expense budget for one year, x million dollars, is comprised of the departmental expenses, y million dollars, and all other expenses, 201 million dollars, the expressions x plus y and 2 x minus y  both must be equivalent to a value greater than 201 million dollars. Therefore, the equations x plus y, equals 201 and 2 x minus y, equals 201 aren’t true. Choice D is incorrect. The value of x must be greater than the value of y. Therefore, y minus x, equals 201 can’t represent this relationship.

 

Question 220 220 of 569 selected Systems Of 2 Linear Equations In 2 Variables M

y=-15x

y=17x

The solution to the given system of equations is (x,y). What is the value of x ?

  1. -5

  2. 0

  3. 2

  4. 7

Show Answer Correct Answer: B

Choice B is correct. It's given by the first equation in the system that y=-15x. Substituting -15x for y in the second equation in the system, y=17x, yields -15x=17x. Adding -15x to both sides of this equation yields 0=17x+15x, which is equivalent to 0=535x+735x, or 0=1235x. Multiplying both sides of this equation by 35 12 yields 0=x. Therefore, the value of x is 0 .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 221 221 of 569 selected Systems Of 2 Linear Equations In 2 Variables E

4 x = 20

- 3 x + y = -7

The solution to the given system of equations is (x,y). What is the value of x + y ?

  1. -27

  2. -13

  3. 13

  4. 27

Show Answer Correct Answer: C

Choice C is correct. It's given that 4 x = 20 and - 3 x + y = -7 is a system of equations with a solution (x,y). Adding the second equation in the given system to the first equation yields 4x+(-3x+y)=20+(-7), which is equivalent to x+y=13. Thus, the value of x+y is 13

Choice A is incorrect. This represents the value of -2(x+y)-1.

Choice B is incorrect. This represents the value of -(x+y).

Choice D is incorrect. This represents the value of 2(x+y)+1.

Question 222 222 of 569 selected Linear Functions E

  • The line slants sharply down from left to right.
  • The line passes through the following points:
    • (0 comma 8)
    • (StartFraction 11 Over 2 EndFraction comma 0)

The graph of the linear function f is shown, where y=f(x). What is the y-intercept of the graph of f?

  1. (0,0)

  2. (0,-1611)

  3. (0,-8)

  4. (0,8)

Show Answer Correct Answer: D

Choice D is correct. The y-intercept of a graph is the point where the graph intersects the y-axis. The graph of function f shown intersects the y-axis at the point (0,8). Therefore, the y-intercept of the graph of f is (0,8).

Choice A is incorrect. This is the point where the x-axis, not the graph of f , intersects the y-axis.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Question 223 223 of 569 selected Linear Inequalities In 1 Or 2 Variables M

A certain elephant weighs 200 pounds at birth and gains more than 2 but less than 3 pounds per day during its first year. Which of the following inequalities represents all possible weights w, in pounds, for the elephant 365 days after birth?

  1. 400 is less than w, which is less than 600

  2. 565 is less than w, which is less than 930

  3. 730 is less than w, which is less than 1,095

  4. 930 is less than w, which is less than 1,295

Show Answer Correct Answer: D

Choice D is correct. It’s given that the elephant weighs 200 pounds at birth and gains more than 2 pounds but less than 3 pounds per day during its first year. The inequality 200 plus 2 d, is less than w, which is less than 200 plus 3 d represents this situation, where d is the number of days after birth. Substituting 365 for d in the inequality gives 200 plus 2, times 365, is less than w, which is less than 200 plus 3, times 365, or 930 is less than w, which is less than 1,295.

Choice A is incorrect and may result from solving the inequality 200 times 2, is less than w, which is less than 200 times 3. Choice B is incorrect and may result from solving the inequality for a weight range of more than 1 pound but less than 2 pounds: 200 plus 1, times 365, is less than w, which is less than 200 plus 2, times 365. Choice C is incorrect and may result from calculating the possible weight gained by the elephant during the first year without adding the 200 pounds the elephant weighed at birth.

 

Question 224 224 of 569 selected Linear Functions M

f of x equals, the fraction with numerator x plus 7, and denominator 4

For the function f defined above, what is the value of f of 9, minus, f of 1 ?

  1. 1

  2. 2

  3. one fourth

  4. nine fourths

Show Answer Correct Answer: B

Choice B is correct. The value of f of 9 minus f of 1 can be calculated by finding the values of f of 9 and f of 1. The value of f of 9 can be found by substituting 9 for x in the given function: f of 9 equals, the fraction with numerator, open parenthesis, 9 plus 7, close parenthesis, and denominator 4. This equation can be rewritten as f  of 9 equals, 16 over 4, or 4. Then, the value of f of 1 can be found by substituting 1 for x in the given function: f of 1 equals, the fraction with numerator, open parenthesis, 1 plus 7, close parenthesis, and denominator 4. This equation can be rewritten as f of 1 equals, 8 over 4, or 2. Therefore, f of 9 minus f of 1, equals, 4 minus 2, which is equivalent to 2.

Choices A, C, and D are incorrect and may result from incorrectly substituting values of x in the given function or making computational errors.

 

Question 225 225 of 569 selected Linear Equations In 1 Variable M

In the equation above, a and b are constants. If the equation has infinitely many solutions, what are the values of a and b ?

  1. a equals 2 and b equals 1

  2. a equals 2 and b equals 7

  3. a equals negative 2 and b equals 5

  4. a equals negative 2 and b equals negative 5

Show Answer Correct Answer: B

Choice B is correct. Distributing the a on the left-hand side of the equation gives 3a – b – ax = –1 – 2x. Rearranging the terms in each side of the equation yields –ax + 3a b = –2x –1. Since the equation has infinitely many solutions, it follows that the coefficients of x and the free terms on both sides must be equal. That is,  –a = –2, or a = 2, and 3ab = –1. Substituting 2 for a in the equation 3a – b = –1 gives 3(2) – b = –1, so  b = 7.

Choice A is incorrect and may be the result of a conceptual error when finding the value of b. Choices C and D are incorrect and may result from making a sign error when simplifying.

Question 226 226 of 569 selected Systems Of 2 Linear Equations In 2 Variables H

A sample of a certain alloy has a total mass of 50.0 grams and is 50.0% silicon by mass. The sample was created by combining two pieces of different alloys. The first piece was 30.0% silicon by mass and the second piece was 80.0% silicon by mass. What was the mass, in grams, of the silicon in the second piece? 

  1. 9.0

  2. 16.0

  3. 20.0

  4. 30.0

Show Answer Correct Answer: B

Choice B is correct. Let x represent the total mass, in grams, of the first piece, and let y represent the total mass, in grams, of the second piece. It's given that the sample has a total mass of 50.0 grams. Therefore, the equation x+y=50.0 represents this situation. It's also given that the sample is 50.0% silicon by mass. Therefore, the total mass of the silicon in the sample is 0.500(50.0), or 25.0, grams. It's also given that the first piece was 30.0% silicon by mass and the second piece was 80.0% silicon by mass. Therefore, the masses, in grams, of the silicon in the first and second pieces can be represented by the expressions 0.300x and 0.800y, respectively. Since the sample was created by combining the first and second pieces, and the total mass of the silicon in the sample is 25.0 grams, the equation 0.300x+0.800y=25.0 represents this situation. Subtracting y from both sides of the equation x+y=50.0 yields x=50.0-y. Substituting 50.0-y for x in the equation 0.300x+0.800y=25.0 yields 0.300(50.0-y)+0.800y=25.0. Distributing 0.300 on the left-hand side of this equation yields 15.0-0.300y+0.800y=25.0. Combining like terms on the left-hand side of this equation yields 15.0+0.500y=25.0. Subtracting 15.0 from both sides of this equation yields 0.500y=10.0. Dividing both sides of this equation by 0.500 yields y=20.0. Substituting 20.0 for y in the expression representing the mass, in grams, of the silicon in the second piece, 0.800y, yields 0.800(20.0), or 16.0. Therefore, the mass, in grams, of the silicon in the second piece is 16.0.

Choice A is incorrect. This is the mass, in grams, of the silicon in the first piece, not the second piece.

Choice C is incorrect. This is the total mass, in grams, of the second piece, not the mass, in grams, of the silicon in the second piece.

Choice D is incorrect. This is the total mass, in grams, of the first piece, not the mass, in grams, of the silicon in the second piece.

Question 227 227 of 569 selected Linear Equations In 1 Variable E

A total of 165 people contributed to a charity event as either a donor or a volunteer. 130 people contributed as a donor. How many people contributed as a volunteer?

  1. 35

  2. 130

  3. 165

  4. 330

Show Answer Correct Answer: A

Choice A is correct. It’s given that a total of 165 people contributed to a charity event as either a donor or a volunteer. It’s also given that 130 people contributed as a donor. It follows that 165-130, or 35 , people contributed as a volunteer.

Choice B is incorrect. This is the number of people who contributed as a donor, not a volunteer.

Choice C is incorrect. This is the total number of people who contributed as either a donor or a volunteer, not the number of people who contributed as a volunteer.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 228 228 of 569 selected Linear Equations In 1 Variable E

4 x + 6 = 18

Which equation has the same solution as the given equation?

  1. 4 x = 108

  2. 4 x = 24

  3. 4 x = 12

  4. 4 x = 3

Show Answer Correct Answer: C

Choice C is correct. Subtracting 6 from both sides of the given equation yields 4 x = 12 , which is the equation given in choice C. Since this equation is equivalent to the given equation, it has the same solution as the given equation.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 229 229 of 569 selected Linear Inequalities In 1 Or 2 Variables M

An event planner is planning a party. It costs the event planner a onetime fee of $35 to rent the venue and $10.25 per attendee. The event planner has a budget of $300 . What is the greatest number of attendees possible without exceeding the budget?

Show Answer Correct Answer: 25

The correct answer is 25 . The total cost of the party is found by adding the onetime fee of the venue to the cost per attendee times the number of attendees. Let x be the number of attendees. The expression 35+10.25x thus represents the total cost of the party. It's given that the budget is $300, so this situation can be represented by the inequality 35+10.25x300. Subtracting 35 from both sides of this inequality gives 10.25x265. Dividing both sides of this inequality by 10.25 results in approximately x25.854. Since the question is stated in terms of attendees, rounding 25.854 down to the greatest whole number gives the greatest number of attendees possible, which is 25 .

Question 230 230 of 569 selected Linear Inequalities In 1 Or 2 Variables E

For a party, 50 dinner rolls are needed. Dinner rolls are sold in packages of 12 . What is the minimum number of packages that should be bought for the party?

Show Answer Correct Answer: 5

The correct answer is 5 . Let p represent the number of packages of dinner rolls that should be bought for the party. It's given that dinner rolls are sold in packages of 12 . Therefore, 12 p represents the number of dinner rolls that should be bought for the party. It's also given that 50 dinner rolls are needed; therefore, 12p50. Dividing both sides of this inequality by 12 yields p5012, or approximately p4.17. Since the number of packages of dinner rolls must be a whole number, the minimum number of packages that should be bought for the party is 5 .

Question 231 231 of 569 selected Linear Equations In 2 Variables H

a, x, plus b y, equals b

In the equation above, a and b are constants and 0 is less than a, which is less than b. Which of the following could represent the graph of the equation in the xy-plane?

  1.  

    The answer choice presents the graph of a line in the x y-plane. The number 1 is indicated on both axes. The line slants downward and to the right, crossing the y axis at 1, and crossing the x axis at 1.

     

  2.  

    The answer choice presents the graph of a line in the x y-plane. The number 1 is indicated on both axes. The line slants downward and to the right, crossing the x axis at negative 1, and crossing the y axis at negative 1.

     

  3.  

    The answer choice presents the graph of a line in the x y-plane. The number 1 is indicated on both axes. The line slants downward and to the right, crossing the y axis at 1, and crossing the x axis at 2.

     

  4.  

    The answer choice presents the graph of a line in the x y-plane. The number 1 is indicated on both axes. The line slants downward and to the right, crossing the y axis at 1, and crossing the x axis at 0 point 5.

     

Show Answer Correct Answer: C

Choice C is correct. The given equation a, x plus b y, equals b can be rewritten in slope-intercept form, y equals, m x plus k, where m represents the slope of the line represented by the equation, and k represents the y-coordinate of the y-intercept of the line. Subtracting ax from both sides of the equation yields b y equals, negative a, x plus b, and dividing both sides of this equation by b yields y equals, the negative of the fraction a over b, end fraction, times x, plus, the fraction b over b, end fraction, or y equals, the negative of the fraction a over b, end fraction, times x, plus 1. With the equation now in slope-intercept form, it shows that k equals 1, which means the y-coordinate of the y-intercept is 1. It’s given that a and b are both greater than 0 (positive) and that a is less than b. Since m equals, the negative of the fraction a over b, the slope of the line must be a value between negative 1 and 0. Choice C is the only graph of a line that has a y-value of the y-intercept that is 1 and a slope that is between negative 1 and 0.

Choices A, B, and D are incorrect because the slopes of the lines in these graphs aren’t between negative 1 and 0.

 

Question 232 232 of 569 selected Linear Inequalities In 1 Or 2 Variables M

H equals, 120 p, plus 60

The Karvonen formula above shows the relationship between Alice’s target heart rate H, in beats per minute (bpm), and the intensity level p of different activities. When p equals 0, Alice has a resting heart rate. When p equals 1, Alice has her maximum heart rate. It is recommended that p be between 0.5 and 0.85 for Alice when she trains. Which of the following inequalities describes Alice’s target training heart rate?

  1. 120 is less than or equal to H, which is less than or equal to 162

  2. 102 is less than or equal to H, which is less than or equal to 120

  3. 60 is less than or equal to H, which is less than or equal to 162

  4. 60 is less than or equal to H, which is less than or equal to 102

Show Answer Correct Answer: A

Choice A is correct. When Alice trains, it’s recommended that p be between 0.5 and 0.85. Therefore, her target training heart rate is represented by the values of H corresponding to 0 point 5 is less than or equal to p, which is less than or equal to 0 point 8 5. When p equals 0 point 5, H equals, 120 times 0 point 5, plus 60, or H equals 120. When p equals 0 point 8 5, H equals, 120 times 0 point 8 5, plus, 60, or H equals 162. Therefore, the inequality that describes Alice’s target training heart rate is 120 is less than or equal to H, which is less than or equal to 162.

Choice B is incorrect. This inequality describes Alice’s target heart rate for 0 point 3 5 is less than or equal to p, which is less than or equal to 0 point 5. Choice C is incorrect. This inequality describes her target heart rate for 0 is less than or equal to p, which is less than or equal to 0 point 8 5. Choice D is incorrect. This inequality describes  her target heart rate for 0 is less than or equal to p, which is less than or equal to 0 point 3 5.

 

Question 233 233 of 569 selected Linear Functions E

The function h is defined by h(x)=3x-7. What is the value of h(-2)?

  1. -13

  2. -10

  3. 10

  4. 13

Show Answer Correct Answer: A

Choice A is correct. The value of h(-2) can be found by substituting -2 for x in the equation defining h . Substituting -2 for x in h(x)=3x-7 yields h(-2)=3(-2)-7, or h(-2)=-13. Therefore, the value of h(-2) is -13 .

Choice B is incorrect. This is the value of h(-1), not h(-2).

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.
 

Question 234 234 of 569 selected Linear Equations In 1 Variable H

2(kx-n)= - 28 15 x - 36 19

In the given equation, k and n are constants and n>1. The equation has no solution. What is the value of k ?

Show Answer Correct Answer: -.9333, -14/15

The correct answer is -1415. A linear equation in the form ax+b=cx+d has no solution only when the coefficients of x on each side of the equation are equal and the constant terms are not equal. Dividing both sides of the given equation by 2 yields kx-n=-2830x-3638, or kx-n=-1415x-1819. Since it’s given that the equation has no solution, the coefficient of x on both sides of this equation must be equal, and the constant terms on both sides of this equation must not be equal. Since 1819<1, and it's given that n>1, the second condition is true. Thus, k must be equal to -1415. Note that -14/15, -.9333, and -0.933 are examples of ways to enter a correct answer.

Question 235 235 of 569 selected Linear Equations In 2 Variables E

y = - 4 x + 40

Which table gives three values of x and their corresponding values of y for the given equation?

Show Answer Correct Answer: C

Choice C is correct. Each of the given choices gives three values of x : 0 , 1 , and 2 . Substituting 0 for x in the given equation yields y=-4(0)+40, or y = 40 . Therefore, when x = 0 , the corresponding value of y for the given equation is 40 . Substituting 1 for x in the given equation yields y=-4(1)+40, or y = 36 . Therefore, when x = 1 , the corresponding value of y for the given equation is 36 . Substituting 2 for x in the given equation yields y=-4(2)+40, or y = 32 . Therefore, when x = 2 , the corresponding value of y for the given equation is 32 . Choice C gives three values of x , 0 , 1 , and 2 , and their corresponding values of y , 40 , 36 , and 32 , respectively, for the given equation.

Choice A is incorrect. This table gives three values of x and their corresponding values of y for the equation y = - 4 x .

Choice B is incorrect. This table gives three values of x and their corresponding values of y for the equation y = 4 x + 40 .

Choice D is incorrect. This table gives three values of x and their corresponding values of y for the equation y = 4 x .

Question 236 236 of 569 selected Linear Equations In 1 Variable M

An agricultural scientist studying the growth of corn plants recorded the height of a corn plant at the beginning of a study and the height of the plant each day for the next 12 days. The scientist found that the height of the plant increased by an average of 1.20 centimeters per day for the 12 days. If the height of the plant on the last day of the study was 36.8 centimeters, what was the height, in centimeters, of the corn plant at the beginning of the study?

Show Answer

The correct answer is 22.4. If the height of the plant increased by an average of 1.20 centimeters per day for 12 days, then its total growth over the 12 days was 1 point 2 0, times 12, equals 14.4 centimeters. The plant was 36.8 centimeters tall after 12 days, so at the beginning of the study its height was 36 point 8, minus 14 point 4, equals 22 point 4 centimeters. Note that 22.4 and 112/5 are examples of ways to enter a correct answer.

Alternate approach: The equation 36 point 8 equals, 12, times, 1 point 2 0, plus h can be used to represent this situation, where h is the height of the plant, in centimeters, at the beginning of the study. Solving this equation for h yields 22.4 centimeters.

 

Question 237 237 of 569 selected Linear Functions E
x f(x)
0 29
1 32
2 35

For the linear function f , the table shows three values of x and their corresponding values of f(x). Which equation defines f(x)?

  1. f(x)= 3 x + 29

  2. f(x)= 29 x + 32

  3. f(x)= 35 x + 29

  4. f(x)= 32 x + 35

Show Answer Correct Answer: A

Choice A is correct. An equation that defines a linear function f can be written in the form f(x)=mx+b, where m and b are constants. It's given in the table that when x = 0 , f(x)=29. Substituting 0 for x and 29 for f(x) in the equation f(x)=mx+b yields 29=m(0)+b, or 29 = b . Substituting 29 for b in the equation f(x)=mx+b yields f(x)=mx+29. It's also given in the table that when x = 1 f(x)=32. Substituting 1 for x and 32 for f(x) in the equation f(x)=mx+29 yields 32=m(1)+29, or 32 = m + 29 . Subtracting 29 from both sides of this equation yields 3 = m . Substituting 3 for m in the equation f(x)=mx+29 yields f(x)=3x+29.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 238 238 of 569 selected Linear Equations In 2 Variables H

The figure presents a scatterplot titled “Number of Cornflowers and Wallflowers at Garden Store.” The horizontal axis is labeled “Number of cornflowers,” and the numbers 0 through 20, in increments of 2, are indicated. The vertical axis is labeled “Number of wallflowers,” and the numbers 0 through 20, in increments of 2, are indicated. There are 5 data points on the graph. The data represented by the points are as follows. 
Data point 1. Number of cornflowers, 0. Number of wallflowers, 12.
Data point 2. Number of cornflowers, 4. Number of wallflowers, 9.
Data point 3. Number of cornflowers, 8. Number of wallflowers, 6.
Data point 4. Number of cornflowers, 12. Number of wallflowers, 3.
Data point 5. Number of cornflowers, 16. Number of wallflowers, 0.

The points plotted in the coordinate plane above represent the possible numbers of wallflowers and cornflowers that someone can buy at the Garden Store in order to spend exactly $24.00 total on the two types of flowers. The price of each wallflower is the same and the price of each cornflower is the same. What is the price, in dollars, of 1 cornflower?

Show Answer

The correct answer is 1.5. The point with coordinates 16 comma 0 corresponds to the situation where 16 cornflowers and 0 wallflowers are purchased. Since the total spent on the two types of flowers is $24.00, it follows that the price of 16 cornflowers is $24.00, and the price of one cornflower is $1.50. Note that 1.5 and 3/2 are examples of ways to enter a correct answer.

Question 239 239 of 569 selected Linear Inequalities In 1 Or 2 Variables H

A small business owner budgets $2,200 to purchase candles. The owner must purchase a minimum of 200 candles to maintain the discounted pricing. If the owner pays $4.90 per candle to purchase small candles and $11.60 per candle to purchase large candles, what is the maximum number of large candles the owner can purchase to stay within the budget and maintain the discounted pricing?

Show Answer Correct Answer: 182

The correct answer is 182 . Let s represent the number of small candles the owner can purchase, and let l represent the number of large candles the owner can purchase. It’s given that the owner pays $4.90 per candle to purchase small candles and $11.60 per candle to purchase large candles. Therefore, the owner pays 4.90s dollars for s small candles and 11.60l dollars for l large candles, which means the owner pays a total of 4.90s+11.60l dollars to purchase candles. It’s given that the owner budgets $2,200 to purchase candles. Therefore, 4.90s+11.60l2,200. It’s also given that the owner must purchase a minimum of 200 candles. Therefore, s+l200. The inequalities 4.90s+11.60l2,200 and s+l200 can be combined into one compound inequality by rewriting the second inequality so that its left-hand side is equivalent to the left-hand side of the first inequality. Subtracting l from both sides of the inequality s+l200 yields s200-l. Multiplying both sides of this inequality by 4.90 yields 4.90s4.90(200-l), or 4.90s980-4.90l. Adding 11.60l to both sides of this inequality yields 4.90s+11.60l980-4.90l+11.60l, or 4.90s+11.60l980+6.70l. This inequality can be combined with the inequality 4.90s+11.60l2,200, which yields the compound inequality 980+6.70l4.90s+11.60l2,200. It follows that 980+6.70l2,200. Subtracting 980 from both sides of this inequality yields 6.70l2,200. Dividing both sides of this inequality by 6.70 yields approximately l182.09. Since the number of large candles the owner purchases must be a whole number, the maximum number of large candles the owner can purchase is the largest whole number less than 182.09 , which is 182 .

Question 240 240 of 569 selected Linear Equations In 1 Variable M

13(x+6)-12(x+6)=-8

What value of x is the solution to the given equation?

Show Answer Correct Answer: 42

The correct answer is 42 . The expression (x+6) is a factor of both terms on the left-hand side of the given equation. Therefore, the given equation can be written as (x+6)(13-12)=-8, or (x+6)(-16)=-8. Multiplying each side of this equation by - 6 yields x+6=48. Subtracting 6 from each side of this equation yields x=42. Therefore, the value of x that is the solution to the given equation is 42 .

Question 241 241 of 569 selected Linear Inequalities In 1 Or 2 Variables M

A number x is at most 17 less than 5 times the value of y . If the value of y is 3 , what is the greatest possible value of x ?

Show Answer Correct Answer: -2

The correct answer is -2 . It's given that a number x is at most 17 less than 5 times the value of y , or x5y-17. Substituting 3 for y in this inequality yields x5(3)-17, or x-2. Thus, if the value of y is 3 , the greatest possible value of x is -2 .

Question 242 242 of 569 selected Linear Equations In 2 Variables E

A food truck buys forks for $0.04 each and plates for $0.48 each. The total cost of x forks and y plates is $661.76. Which equation represents this situation?

  1. 0.48x-0.04y=661.76

  2. 0.04x-0.48y=661.76

  3. 0.48x+0.04y=661.76

  4. 0.04x+0.48y=661.76

Show Answer Correct Answer: D

Choice D is correct. It’s given that the food truck buys forks for $0.04 each. Therefore, the cost, in dollars, of x forks can be represented by the expression 0.04 x . It’s also given that the food truck buys plates for $0.48 each. Therefore, the cost, in dollars, of y plates can be represented by the expression 0.48 y . Since the total cost of x forks and y plates is $661.76, the equation 0.04 x + 0.48 y = 661.76 represents this situation.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect. This equation represents a situation in which the food truck buys forks for $0.48 each and plates for $0.04 each.

Question 243 243 of 569 selected Linear Equations In 2 Variables H
x y
- 2 s 24
- s 21
s 15

The table shows three values of x and their corresponding values of y , where s is a constant. There is a linear relationship between x and y . Which of the following equations represents this relationship?

  1. sx+3y=18s

  2. 3x+sy=18s

  3. 3x+sy=18

  4. sx+3y=18

Show Answer Correct Answer: B

Choice B is correct. The linear relationship between x and y can be represented by an equation of the form y-y1=m(x-x1), where m is the slope of the graph of the equation in the xy-plane and (x1,y1) is a point on the graph. The slope of a line can be found using two points on the line and the slope formula m=y2-y1x2-x1. Each value of x and its corresponding value of y in the table can be represented by a point (x,y). Substituting the points (-s,21) and (s,15) for (x1,y1) and (x2,y2), respectively, in the slope formula yields m=15-21s-(-s), which gives m=-62s, or m=-3s. Substituting -3s for m  and the point (s,15) for (x1,y1) in the equation y-y1=m(x-x1) yields y-15=-3s(x-s). Distributing -3s on the right-hand side of this equation yields y-15=-3xs+3. Adding 15 to each side of this equation yields y=-3xs+18. Multiplying each side of this equation by s yields sy=-3x+18s. Adding 3 x to each side of this equation yields 3x+sy=18s. Therefore, the equation 3x+sy=18s represents this relationship.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 244 244 of 569 selected Linear Equations In 2 Variables H

The graph of 7 x + 2 y = -31 in the x y -plane has an x -intercept at (a , 0 ) and a y -intercept at (0 , b ), where a and b are constants. What is the value of b a ?

  1. - 7 2

  2. - 2 7

  3. 2 7

  4. 7 2

Show Answer Correct Answer: D

Choice D is correct. The x-coordinate a of the x-intercept (a,0) can be found by substituting 0 for y in the given equation, which gives 7x+2(0)=-31, or 7x=-31. Dividing both sides of this equation by 7 yields x=-317. Therefore, the value of a is -317. The y-coordinate b of the y-intercept (0,b) can be found by substituting 0 for x in the given equation, which gives 7(0)+2y=-31, or 2y=-31. Dividing both sides of this equation by 2 yields y=-312. Therefore, the value of b is -312. It follows that the value of b a is -312-317, which is equivalent to (312)(731), or 7 2 .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Question 245 245 of 569 selected Linear Inequalities In 1 Or 2 Variables H

A psychologist set up an experiment to study the tendency of a person to select the first item when presented with a series of items. In the experiment, 300 people were presented with a set of five pictures arranged in random order. Each person was asked to choose the most appealing picture. Of the first 150 participants, 36 chose the first picture in the set. Among the remaining 150 participants, p people chose the first picture in the set. If more than 20% of all participants chose the first picture in the set, which of the following inequalities best describes the possible values of p ?

  1. p is greater than, 0 point 20 times, open parenthesis, 300 minus 36, close parenthesis, where p is less than or equal to 150

  2. p is greater than, 0 point 20 times, open parenthesis, 300 plus 36, close parenthesis, where p is less than or equal to 150

  3. p minus 36 is greater than, 0 point 20 times 300, where p is less than or equal to 150

  4. p plus 36 is greater than, 0 point 20 times 300, where p is less than or equal to 150

Show Answer Correct Answer: D

Choice D is correct. Of the first 150 participants, 36 chose the first picture in the set, and of the 150 remaining participants, p chose the first picture in the set. Hence, the proportion of the participants who chose the first picture in the set is the fraction with numerator 36 plus p, and denominator 300. Since more than 20% of all the participants chose the first picture, it follows that the fraction with numerator 36 plus p, and denominator 300, is greater than 0 point 2 0.

This inequality can be rewritten as p plus 36, is greater than, 0 point 2 0 times 300. Since p is a number of people among the remaining 150 participants, p is less than or equal to 150.

Choices A, B, and C are incorrect and may be the result of some incorrect interpretations of the given information or of computational errors.

 

Question 246 246 of 569 selected Systems Of 2 Linear Equations In 2 Variables M

The score on a trivia game is obtained by subtracting the number of incorrect answers from twice the number of correct answers. If a player answered 40 questions and obtained a score of 50, how many questions did the player answer correctly?

Show Answer

The correct answer is 30. Let x represent the number of correct answers from the player and y represent the number of incorrect answers from the player. Since the player answered 40 questions in total, the equation x plus y, equals 40 represents this situation. Also, since the score is found by subtracting the number of incorrect answers from twice the number of correct answers and the player received a score of 50, the equation 2 x minus y, equals 50 represents this situation. Adding the equations in the system of two equations together yields open parenthesis, x plus y, close parenthesis, plus, open parenthesis, 2 x minus y, close parenthesis, equals, 40 plus 50. This can be rewritten as 3 x equals 90. Finally, solving for x by dividing both sides of the equation by 3 yields x equals 30.

Question 247 247 of 569 selected Linear Equations In 1 Variable E

Nasir bought 9 storage bins that were each the same price. He used a coupon for $63 off the entire purchase. The cost for the entire purchase after using the coupon was $27. What was the original price, in dollars, for 1 storage bin?

Show Answer Correct Answer: 10

The correct answer is 10 . It’s given that the cost for the entire purchase was $27 after a coupon was used for $63 off the entire purchase. Adding the amount of the coupon to the purchase price yields 27+63=90. Thus, the cost for the entire purchase before using the coupon was $90. It’s given that Nasir bought 9 storage bins. The original price for 1 storage bin can be found by dividing the total cost by 9 . Therefore, the original price, in dollars, for 1 storage bin is 909, or 10 .

Question 248 248 of 569 selected Linear Equations In 2 Variables M

  • The line slants down from left to right.
  • The line passes through the following points:
    • (0 comma 40)
    • (60 comma 0)

The graph shows the relationship between the number of shares of stock from Company A, x , and the number of shares of stock from Company B, y , that Simone can purchase. Which equation could represent this relationship?

  1. y = 8 x + 12

  2. 8 x + 12 y = 480

  3. y = 12 x + 8

  4. 12 x + 8 y = 480

Show Answer Correct Answer: B

Choice B is correct. The graph shown is a line passing through the points (0,40) and (60,0). Since the relationship between x and y is linear, if two points on the graph make a linear equation true, then the equation represents the relationship. Substituting 0 for x and 40 for y in the equation in choice B, 8 x + 12 y = 480 , yields 8(0)+12(40)=480, or 480=480, which is true. Substituting 60 for x and 0 for y in the equation 8 x + 12 y = 480 yields 8(60)+12(0)=480, or 480=480, which is true. Therefore, the equation 8 x + 12 y = 480 represents the relationship between x and y .

Choice A is incorrect. The point (0,40) is not on the graph of this equation, since 40=8(0)+12, or 40=12, is not true.

Choice C is incorrect. The point (0,40) is not on the graph of this equation, since 40=12(0)+8, or 40=8, is not true.

Choice D is incorrect. The point (0,40) is not on the graph of this equation, since 12(0)+8(40)=480, or 320=480, is not true.

Question 249 249 of 569 selected Linear Equations In 2 Variables E

4 x plus 3 y, equals 24

Mario purchased 4 binders that cost x dollars each and 3 notebooks that cost y dollars each. If the given equation represents this situation, which of the following is the best interpretation of 24 in this context?

  1. The total cost, in dollars, for all binders purchased

  2. The total cost, in dollars, for all notebooks purchased

  3. The total cost, in dollars, for all binders and notebooks purchased

  4. The difference in the total cost, in dollars, between the number of binders and notebooks purchased

Show Answer Correct Answer: C

Choice C is correct. Since Mario purchased 4 binders that cost x dollars each, the expression 4 x represents the total cost, in dollars, of the 4 binders he purchased. Since Mario purchased 3 notebooks that cost y dollars each, the expression 3 y represents the total cost, in dollars, of the 3 notebooks he purchased. Therefore, the expression 4 x plus 3 y represents the total cost, in dollars, for all binders and notebooks he purchased. In the given equation, the expression 4 x plus 3 y is equal to 24. Therefore, it follows that 24 is the total cost, in dollars, for all binders and notebooks purchased.

Choice A is incorrect. This is represented by the expression 4 x in the given equation. Choice B is incorrect. This is represented by the expression 3 y in the given equation. Choice D is incorrect. This is represented by the expression the absolute value of, 4 x minus 3 y, end absolute value.

 

Question 250 250 of 569 selected Systems Of 2 Linear Equations In 2 Variables H

48 x - 72 y = 30 y + 24

ry=16-16x

In the given system of equations, r is a constant. If the system has no solution, what is the value of r ?

Show Answer Correct Answer: -34

The correct answer is -34. A system of two linear equations in two variables, x and y , has no solution if the lines represented by the equations in the xy-plane are distinct and parallel. Two lines represented by equations in standard form Ax+By=C, where A , B , and C are constants, are parallel if the coefficients for x and y in one equation are proportional to the corresponding coefficients in the other equation. The first equation in the given system can be written in standard form by subtracting 30 y from both sides of the equation to yield 48x-102y=24. The second equation in the given system can be written in standard form by adding 16x to both sides of the equation to yield 16x+ry=16.  The coefficient of x in this second equation, 16 , is 13 times the coefficient of x in the first equation, 48 . For the lines to be parallel the coefficient of y in the second equation, r , must also be 13 times the coefficient of y in the first equation, -102. Thus, r=13(-102), or r=-34. Therefore, if the given system has no solution, the value of r is -34.

Question 251 251 of 569 selected Linear Equations In 1 Variable E

10 equals, 2 x plus 4

How many solutions exist to the equation shown above?

  1. None

  2. Exactly 1

  3. Exactly 3

  4. Infinitely many

Show Answer Correct Answer: B

Choice B is correct. Subtracting 4 from each side of the given equation yields 6 equals 2 x, or x equals 3, so the equation has a unique solution of x equals 3.

Choice A is incorrect. Since 3 is a value of x that satisfies the given equation, the equation has at least 1 solution. Choice C is incorrect. Linear equations can have 0, 1, or infinitely many solutions; no linear equation has exactly 3 solutions. Choice D is incorrect. If a linear equation has infinitely many solutions, it can be reduced to 0 equals 0. This equation reduces to x equals 3, so there is only 1 solution.

 

Question 252 252 of 569 selected Linear Functions E

For a training program, Juan rides his bike at an average rate of 5.7 minutes per mile. Which function m models the number of minutes it will take Juan to ride x miles at this rate?

  1. m(x)=x5.7

  2. m(x)=x+5.7

  3. m(x)=x-5.7

  4. m(x)=5.7x

Show Answer Correct Answer: D

Choice D is correct. It′s given that Juan rides his bike at an average rate of 5.7 minutes per mile. The number of minutes it will take Juan to ride x miles can be determined by multiplying his average rate by the number of miles, x , which yields 5.7x. Therefore, the function m(x)=5.7x models the number of minutes it will take Juan to ride x miles.

Choice A is incorrect and may result from conceptual errors. 

Choice B is incorrect and may result from conceptual errors. 

Choice C is incorrect and may result from conceptual errors. 

Question 253 253 of 569 selected Linear Functions H
x f(x)
1 -64
2 0
3 64

For the linear function f , the table shows three values of x and their corresponding values of f(x). Function f is defined by f(x)=ax+b, where a and b are constants. What is the value of a - b ?

  1. -64

  2. 62

  3. 128

  4. 192

Show Answer Correct Answer: D

Choice D is correct. The table gives that f(x)=0 when x=2. Substituting 0 for f(x) and 2 for x into the equation f(x)=ax+b yields 0=2a+b. Subtracting 2 a from both sides of this equation yields b=-2a. The table gives that f(x)=-64 when x=1. Substituting -2a for b , -64 for f(x), and 1 for x  into the equation f(x)=ax+b yields -64=a(1)+(-2a). Combining like terms yields -64=-a, or a=64. Since b=-2a, substituting 64 for a into this equation gives b=(-2)(64), which yields b=-128. Thus, the value of a-b can be written as 64-(-128), which is 192 .

Choice A is incorrect. This is the value of a+b, not a-b.

Choice B is incorrect. This is the value of a-2, not a-b.

Choice C is incorrect. This is the value of 2a, not a-b.

Question 254 254 of 569 selected Linear Equations In 1 Variable E

x + 40 = 95

What value of x is the solution to the given equation?

Show Answer Correct Answer: 55

The correct answer is 55 . Subtracting 40 from both sides of the given equation yields x=55. Therefore, the value of x is 55 .

Question 255 255 of 569 selected Systems Of 2 Linear Equations In 2 Variables M

y-9x=13

5x=2y

What is the solution (x,y) to the given system of equations? 

  1. (52,1)

  2. (1,25)

  3. (-2,-5)

  4. (-5,-2)

Show Answer Correct Answer: C

Choice C is correct. Adding 9 x to both sides of the first equation in the given system yields y = 9 x + 13 . Substituting the expression 9 x + 13 for y in the second equation in the given system yields 5x=2(9x+13). Distributing the 2 on the right-hand side of this equation yields 5 x = 18 x + 26 . Subtracting 18 x from both sides of this equation yields - 13 x = 26 . Dividing both sides of this equation by -13 yields x = -2 . Substituting -2 for x in the equation y = 9 x + 13 yields y=9(-2)+13, or y = -5 . Therefore, the solution (x,y) to the given system of equations is (-2,-5).

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect. This is the solution (y,x), not (x,y), to the given system of equations.

Question 256 256 of 569 selected Systems Of 2 Linear Equations In 2 Variables E

y = 4 x - 9
y = 19

What is the solution (x,y) to the given system of equations?

  1. (4,19)

  2. (7,19)

  3. (19,4)

  4. (19,7)

Show Answer Correct Answer: B

Choice B is correct. It's given by the second equation in the system that y = 19 . Substituting 19 for y in the first equation yields 19 = 4 x - 9 . Adding 9 to both sides of this equation yields 28 = 4 x . Dividing both sides of this equation by 4 yields 7 = x . Therefore, since x = 7 and y = 19 , the solution (x,y) to the given system of equations is (7,19).

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 257 257 of 569 selected Linear Functions H

Oil and gas production in a certain area dropped from 4 million barrels in 2000 to 1.9 million barrels in 2013. Assuming that the oil and gas production decreased at a constant rate, which of the following linear functions f best models the production, in millions of barrels, t years after the year 2000?

  1. f of t equals, the fraction 21 over 130, end fraction, t plus 4

  2. f of t equals, the fraction 19 over 130, end fraction, t plus 4

  3. f of t equals, the negative fraction 21 over 130, end fraction, t plus 4

  4. f of t equals, the negative fraction 19 over 130, end fraction, t plus 4

Show Answer Correct Answer: C

Choice C is correct. It is assumed that the oil and gas production decreased at a constant rate. Therefore, the function f that best models the production t years after the year 2000 can be written as a linear function, f of t equals, m, t plus b, where m is the rate of change of the oil and gas production and b is the oil and gas production, in millions of barrels, in the year 2000. Since there were 4 million barrels of oil and gas produced in 2000, b equals 4. The rate of change, m, can be calculated as the fraction with numerator 4 minus 1 point 9, and denominator 0 minus 13, end fraction, equals, the negative of the fraction 2 point 1 over 13, which is equivalent to the negative of the fraction 21 over 130, the rate of change in choice C.

Choices A and B are incorrect because each of these functions has a positive rate of change. Since the oil and gas production decreased over time, the rate of change must be negative. Choice D is incorrect. This model may result from misinterpreting 1.9 million barrels as the amount by which the production decreased.

 

Question 258 258 of 569 selected Linear Functions M
The figure presents a 2-column table, with 3 rows of data. The heading for the first column is “x,” and the heading for the second column is “f of x.” The 3 rows of data are as follows. 
Row 1. x, zero; f of x, negative 2. 
Row 2. x, 2; f of x, 4.
Row 3. x, 6; f of x, 16.

Some values of the linear function f are shown in the table above. What is the value of f of 3?

  1. 6

  2. 7

  3. 8

  4. 9

Show Answer Correct Answer: B

Choice B is correct. A linear function has a constant rate of change, and any two rows of the table shown can be used to calculate this rate. From the first row to the second, the value of x is increased by 2 and the value of f of x is increased by 6 equals, 4 minus negative 2. So the values of f of x increase by 3 for every increase by 1 in the value of x. Since f of 2, equals 4, it follows that f of, open parenthesis, 2 plus 1, close parenthesis, equals, 4 plus 3, which equals 7. Therefore, f of 3, equals 7.

Choice A is incorrect. This is the third x-value in the table, not f of 3. Choices C and D are incorrect and may result from errors when calculating the function’s rate of change.

Question 259 259 of 569 selected Linear Inequalities In 1 Or 2 Variables M

y<x

x<22

For which of the following tables are all the values of x and their corresponding values of y solutions to the given system of inequalities?

  1. x y
    19 18
    20 19
    21 20
  2. x y
    19 20
    20 21
    21 22
  3. x y
    23 22
    24 23
    25 24
  4. x y
    23 24
    24 25
    25 26
Show Answer Correct Answer: A

Choice A is correct. The inequality y<x indicates that for any solution to the given system of inequalities, the value of x must be greater than the corresponding value of y . The inequality x<22 indicates that for any solution to the given system of inequalities, the value of x must be less than 22 . Of the given choices, only choice A contains values of x that are each greater than the corresponding value of y and less than 22 . Therefore, for choice A, all the values of x and their corresponding values of y are solutions to the given system of inequalities.

Choice B is incorrect. The values in this table aren’t solutions to the inequality y<x.

Choice C is incorrect. The values in this table aren’t solutions to the inequality x<22.

Choice D is incorrect. The values in this table aren’t solutions to the inequality y<x or the inequality x<22.

Question 260 260 of 569 selected Linear Equations In 1 Variable M

A line segment that has a length of 115 centimeters (cm) is divided into three parts. One part is 47 cm long. The other two parts have lengths that are equal to each other. What is the length, in cm, of one of the other two parts of equal length?

Show Answer Correct Answer: 34

The correct answer is 34 . It’s given that a line segment has a length of 115 cm and is divided into three parts, where one part is 47 cm long and the other two parts have lengths that are equal. If x represents the length, in cm, of each of the two parts of equal length, then the equation 47+x+x=115, or 47+2x=115, represents this situation. Subtracting 47 from each side of this equation yields 2x=68. Dividing each side of this equation by 2 yields x = 34 . Therefore, the length, in cm, of one of the two parts of equal length is 34 .

Question 261 261 of 569 selected Systems Of 2 Linear Equations In 2 Variables H

24 x + y = 48

6 x + y = 72

The solution to the given system of equations is (x,y). What is the value of y ?

Show Answer Correct Answer: 80

The correct answer is 80 . Subtracting the second equation in the given system from the first equation yields (24x+y)-(6x+y)=48-72, which is equivalent to 24x-6x+y-y=-24, or 18x=-24. Dividing each side of this equation by 3 yields 6x=-8. Substituting -8 for 6 x in the second equation yields -8+y=72. Adding 8 to both sides of this equation yields y=80

Alternate approach: Multiplying each side of the second equation in the given system by 4 yields 24x+4y=288. Subtracting the first equation in the given system from this equation yields (24x+4y)-(24x+y)=288-48, which is equivalent to 24x-24x+4y-y=240, or 3y=240. Dividing each side of this equation by 3 yields y=80.

Question 262 262 of 569 selected Linear Equations In 1 Variable E

If 2 x = 12 , what is the value of 9 x ?

Show Answer Correct Answer: 54

The correct answer is 54 . Dividing both sides of the given equation by 2 yields x=6. Multiplying both sides of this equation by 9 yields 9x=54. Thus, the value of 9x is 54 .

Question 263 263 of 569 selected Linear Functions M

In the xy-plane, the points with coordinates negative 2 comma 3 and 4 comma negative 5 lie on the graph of which of the following linear functions?

  1. f of x equals, x plus 5

  2. f of x equals, one half x, plus 4

  3. f of x equals, negative, four thirds x, plus one third

  4. f of x equals, negative, three halves x, plus 1

Show Answer Correct Answer: C

Choice C is correct. A linear function can be written in the form f of x  equals, m x plus b, where m is the slope and b is the y-coordinate of the y-intercept of the line. The slope of the graph can be found using the formula m equals, the fraction with numerator y sub 2 minus y sub 1, and denominator x sub 2 minus x sub 1, end fraction. Substituting the values of the given points into this formula yields m equals, the fraction with numerator negative 5 minus 3, and denominator 4 minus negative 2, end fraction or m equals, negative 8 over 6, which simplifies to m equals, negative four thirds. Only choice C shows an equation with this slope.

Choices A, B, and D are incorrect and may result from computation errors or misinterpreting the given information.

Question 264 264 of 569 selected Linear Functions E

The function f is defined by f(x)=12(x+6). What is the value of f(4)?

  1. 20

  2. 12

  3. 10

  4. 5

Show Answer Correct Answer: D

Choice D is correct. It’s given that the function f is defined by f(x)=12(x+6). Substituting 4 for x in the given function yields f(4)=12(4+6), or f(4)=5. Therefore, the value of f(4) is 5 .

Choice A is incorrect. This is the value of 2(4+6), not 12(4+6).

Choice B is incorrect. This is the value of 2+(4+6), not 12(4+6).

Choice C is incorrect. This is the value of 4+6, not 12(4+6).

Question 265 265 of 569 selected Linear Inequalities In 1 Or 2 Variables M

y>7x-4

For which of the following tables are all the values of x and their corresponding values of y solutions to the given inequality?

  1. x y
    3 13
    5 27
    8 48
  2. x y
    3 17
    5 31
    8 52
  3. x y
    3 21
    5 27
    8 52
  4. x y
    3 21
    5 35
    8 56
Show Answer Correct Answer: D

Choice D is correct. A solution (x,y) to the given inequality is a value of x and the corresponding value of y such that the value of y is greater than the value of 7x-4. All the tables in the choices have the same three values of x, so each of the three values of x can be substituted in the given inequality to compare the corresponding values of y in each of the tables. Substituting 3 for x in the given inequality yields y>7(3)-4, or y>17. Substituting 5 for x in the given inequality yields y>7(5)-4, or y>31. Substituting 8 for x in the given inequality yields y>7(8)-4, or y>52. Therefore, when x=3, x=5, and x=8, the corresponding values of y must be greater than 17, greater than 31, and greater than 52, respectively. In the table in choice D, when x=3, the corresponding value of y is 21, which is greater than 17; when x=5, the corresponding value of y is 35, which is greater than 31; when x=8, the corresponding value of y is 56, which is greater than 52. Of the given choices, only choice D gives values of x and their corresponding values of y that are all solutions to the given inequality.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Question 266 266 of 569 selected Systems Of 2 Linear Equations In 2 Variables H

A movie theater sells two types of tickets, adult tickets for $12 and child tickets for $8. If the theater sold 30 tickets for a total of $300, how much, in dollars, was spent on adult tickets? (Disregard the $ sign when gridding your answer.)

Show Answer

The correct answer is 180. Let a be the number of adult tickets sold and c be the number of child tickets sold. Since the theater sold a total of 30 tickets, a + c = 30. The price per adult ticket is $12, and the price per child ticket is $8. Since the theater received a total of $300 for the 30 tickets sold, it follows that 12a + 8c = 300. To eliminate c, the first equation can be multiplied by 8 and then subtracted from the second equation:
  12 a, plus 8 c, equals 300, and negative 8 a, minus 8 c, equals negative 240, gives 4 a, plus 0 c, equals 60

Because the question asks for the amount spent on adult tickets, which is 12a dollars, the resulting equation can be multiplied by 3 to give 3(4a) = 3(60) = 180. Therefore, $180 was spent on adult tickets.

Alternate approach: If all the 30 tickets sold were child tickets, their total price would be 30($8) = $240. Since the actual total price of the 30 tickets was $300, the extra $60 indicates that a certain number of adult tickets, a, were sold. Since the price of each adult ticket is $4 more than each child ticket, 4a = 60, and it follows that 12a = 180.

Question 267 267 of 569 selected Linear Functions M
The figure presents the graph of a line in the coordinate plane. The horizontal m-axis is labeled “Time, in minutes,” and the numbers 0 through 7 are indicated. The vertical d-axis is labeled “Distance traveled, in feet,” and the numbers 0 through 7 are indicated. The line passes through the points with coordinates 1 comma 2 and 3 comma 6.

The graph above shows the distance traveled d, in feet, by a product on a conveyor belt m minutes after the product is placed on the belt. Which of the following equations correctly relates d and m ?

  1. d equals 2 m

  2. d equals, one-half m

  3. d equals m, plus 2

  4. d equals 2 m, plus 2

Show Answer Correct Answer: A

Choice A is correct. The line passes through the origin. Therefore, this is a relationship of the form d equals, k m, where k is a constant representing the slope of the graph. To find the value of k, choose a point with coordinates m comma d on the graph of the line other than the origin and substitute the values of m and d into the equation. For example, if the point with coordinates 2 comma 4 is chosen, then 4 equals, k times 2, and k equals 2. Therefore, the equation of the line is d equals, 2 m.

Choice B is incorrect and may result from calculating the slope of the line as the change in time over the change in distance traveled instead of the change in distance traveled over the change in time. Choices C and D are incorrect because each of these equations represents a line with a d-intercept of 2. However, the graph shows a line with a d-intercept of 0.

 

Question 268 268 of 569 selected Linear Equations In 1 Variable H

5(t+3)-7(t+3)=38

What value of t is the solution to the given equation?

Show Answer Correct Answer: -22

The correct answer is -22 . The given equation can be rewritten as -2(t+3)=38. Dividing both sides of this equation by -2 yields t+3=-19. Subtracting 3 from both sides of this equation yields t = -22 . Therefore, -22 is the value of t that is the solution to the given equation.

Question 269 269 of 569 selected Linear Equations In 2 Variables M

Lily made 36 cups of jam. Lily then filled x small containers and y large containers with all the jam she made. The equation 4 x + 6 y = 36 represents this situation. Which is the best interpretation of 6 y in this context?

  1. The number of large containers Lily filled

  2. The number of small containers Lily filled

  3. The total number of cups of jam in the large containers

  4. The total number of cups of jam in the small containers

Show Answer Correct Answer: C

Choice C is correct. It’s given that the equation 4x+6y=36 represents the situation where Lily filled x small containers and y large containers with all the jam she made, which was 36 cups. Therefore, 6y represents the total number of cups of jam in the large containers.

Choice A is incorrect. The number of large containers Lily filled is represented by y, not 6y.

Choice B is incorrect. The number of small containers Lily filled is represented by x, not 6y.

Choice D is incorrect. The total number of cups of jam in the small containers is represented by 4x, not 6y.

Question 270 270 of 569 selected Linear Equations In 1 Variable H

How many solutions does the equation 12(x-3)=-3(x+12) have?

  1. Exactly one

  2. Exactly two

  3. Infinitely many

  4. Zero

Show Answer Correct Answer: A

Choice A is correct. Distributing 12 on the left-hand side and -3 on the right-hand side of the given equation yields 12x-36=-3x-36. Adding 3 x to each side of this equation yields 15x-36=-36. Adding 36 to each side of this equation yields 15 x = 0 . Dividing each side of this equation by 15 yields x = 0 . This means that 0 is the only solution to the given equation. Therefore, the given equation has exactly one solution.

Choice B is incorrect and may result from conceptual or calculation errors. 

Choice C is incorrect and may result from conceptual or calculation errors. 

Choice D is incorrect and may result from conceptual or calculation errors. 

Question 271 271 of 569 selected Linear Functions E

The function g is defined by g(x)=4x-6. What is the value of g(-7)?

  1. -34

  2. -22

  3. - 13 4

  4. - 1 4

Show Answer Correct Answer: A

Choice A is correct. It’s given that the function g is defined by g(x)=4x-6. Substituting - 7 for x into the given equation yields g(-7)=4(-7)-6, or g(-7)=-34.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect. This is the value of x for which g(x)=-7, not the value of g(-7).

Question 272 272 of 569 selected Linear Equations In 2 Variables E

A producer is creating a video with a length of 70 minutes. The video will consist of segments that are 1 minute long and segments that are 3 minutes long. Which equation represents this situation, where x represents the number of 1 -minute segments and y represents the number of 3 -minute segments?

  1. 4 x y = 70

  2. 4(x+y)=70

  3. 3 x + y = 70

  4. x + 3 y = 70

Show Answer Correct Answer: D

Choice D is correct. Since x represents the number of 1 -minute segments and y represents the number of 3 -minute segments, the total length of the video is 1·x+3·y, or x+3y, minutes. Since the video is 70 minutes long, the equation x+3y=70 represents this situation.

Choice A is incorrect and may result from conceptual errors.

Choice B is incorrect and may result from conceptual errors.

Choice C is incorrect and may result from conceptual errors.

Question 273 273 of 569 selected Linear Functions M

c of x equals, m x plus 500

A company’s total cost c of x, in dollars, to produce x shirts is given by the function above, where m is a constant and x is greater than 0. The total cost to produce 100 shirts is $800. What is the total cost, in dollars, to produce 1000 shirts? (Disregard the $ sign when gridding your answer.)

Show Answer

The correct answer is 3500. The given information includes a cost, $800, to produce 100 shirts. Substituting c of x equals 800 and x equals 100 into the given equation yields 800 equals, m times 100, plus 500 . Subtracting 500 from both sides of the equation yields 300 equals, m times 100. Dividing both sides of this equation by 100 yields 3 equals m. Substituting the value of m into the given equation yields c of x equals, 3 x plus 500. Substituting 1000 for x in this equation and solving for c of x gives the cost of 1000 shirts: 3 times 1000, plus 500 , or 3500.

Question 274 274 of 569 selected Linear Functions E

For the linear function f , the graph of y=f(x) in the xy-plane has a slope of 39 and passes through the point (0,0). Which equation defines f ?

  1. f(x)=-39x

  2. f(x)=139x

  3. f(x)=x-39

  4. f(x)=39x

Show Answer Correct Answer: D

Choice D is correct. An equation defining a linear function can be written in the form f(x)=mx+b, where m is the slope and (0,b) is the y-intercept of the graph of y=f(x) in the xy-plane. It’s given that the graph of y=f(x) has a slope of 39 , so m = 39 . It’s also given that the graph of y=f(x) passes through the point (0,0), so b = 0 . Substituting 39 for m and 0 for b in f(x)=mx+b yields f(x)=39x+0, or f(x)=39x. Thus, the equation that defines f is f(x)=39x.

Choice A is incorrect. This equation defines a function whose graph has a slope of -39 , not 39 .

Choice B is incorrect. This equation defines a function whose graph has a slope of 1 39 , not 39 .

Choice C is incorrect. This equation defines a function whose graph has a slope of 1 , not 39 , and passes through the point (0,-39), not (0,0).

Question 275 275 of 569 selected Linear Equations In 2 Variables E

A gardener buys two kinds of fertilizer. Fertilizer A contains 60% filler materials by weight and Fertilizer B contains 40% filler materials by weight. Together, the fertilizers bought by the gardener contain a total of 240 pounds of filler materials. Which equation models this relationship, where x is the number of pounds of Fertilizer A and y is the number of pounds of Fertilizer B?

  1. zero point 4 x, plus zero point 6 y, equals 240

  2. zero point 6 x, plus zero point 4 y, equals 240

  3. 40 x, plus, 60 y, equals 240

  4. 60 x, plus 40 y, equals 240

Show Answer Correct Answer: B

Choice B is correct. Since Fertilizer A contains 60% filler materials by weight, it follows that x pounds of Fertilizer A consists of 0.6x pounds of filler materials. Similarly, y pounds of Fertilizer B consists of 0.4y pounds of filler materials. When x pounds of Fertilizer A and y pounds of Fertilizer B are combined, the result is 240 pounds of filler materials. Therefore, the total amount, in pounds, of filler materials in a mixture of x pounds of Fertilizer A and y pounds of Fertilizer B can be expressed as 0 point 6 x, plus 0 point 4 y, equals 240.

Choice A is incorrect. This choice transposes the percentages of filler materials for Fertilizer A and Fertilizer B. Fertilizer A consists of 0.6x pounds of filler materials and Fertilizer B consists of 0.4y pounds of filler materials. Therefore, 0 point 6 x, plus 0 point 4 y is equal to 240, not 0 point 4 x, plus 0 point 6 y. Choice C is incorrect. This choice transposes the percentages of filler materials for Fertilizer A and Fertilizer B and incorrectly represents how to take the percentage of a value mathematically. Choice D is incorrect. This choice incorrectly represents how to take the percentage of a value mathematically. Fertilizer A consists of 0.6x pounds of filler materials, not 60x pounds of filler materials, and Fertilizer B consists of 0.4y pounds of filler materials, not 40y pounds of filler materials.

 

Question 276 276 of 569 selected Linear Functions H

One gallon of paint will cover 220 square feet of a surface. A room has a total wall area of w  square feet. Which equation represents the total amount of paint P , in gallons, needed to paint the walls of the room twice?

  1. P = w 110

  2. P = 440 w

  3. P = w 220

  4. P = 220 w

Show Answer Correct Answer: A

Choice A is correct. It's given that w represents the total wall area, in square feet. Since the walls of the room will be painted twice, the amount of paint, in gallons, needs to cover 2 w square feet. It’s also given that one gallon of paint will cover 220 square feet. Dividing the total area, in square feet, of the surface to be painted by the number of square feet covered by one gallon of paint gives the number of gallons of paint that will be needed. Dividing 2 w by 220 yields  2w220, or w110. Therefore, the equation that represents the total amount of paint P , in gallons, needed to paint the walls of the room twice is P=w110.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from finding the amount of paint needed to paint the walls once rather than twice.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 277 277 of 569 selected Linear Inequalities In 1 Or 2 Variables M

  • The boundary of the inequality is a dashed line.
    • The line slants sharply down from left to right.
    • The line passes through the following points:
      • (negative 1 comma 2)
      • (0 comma negative 1)
  • The area above and to the right of the boundary is shaded.

The shaded region shown represents the solutions to which inequality?

  1. y<-1+3x

  2. y<-1-3x

  3. y>-1+3x

  4. y>-1-3x

Show Answer Correct Answer: D

Choice D is correct. The equation for the line representing the boundary of the shaded region can be written in slope-intercept form y=b+mx, where m is the slope and (0,b) is the y-intercept of the line. For the graph shown, the boundary line passes through the points (0,-1) and (1,-4). Given two points on a line, (x1,y1) and (x2,y2), the slope of the line can be calculated using the equation m=y2-y1x2-x1. Substituting the points (0,-1) and (1,-4) for (x1,y1) and (x2,y2) in this equation yields m=-4-(-1)1-0, which is equivalent to m=-31, or m=-3. Since the point (0,-1) represents the y-intercept, it follows that b = -1 . Substituting -3 for m and -1 for b in the equation y=b+mx yields y=-1-3x as the equation of the boundary line. Since the shaded region represents all the points above this boundary line, it follows that the shaded region shown represents the solutions to the inequality y>-1-3x.

Choice A is incorrect. This inequality represents a region below, not above, a boundary line with a slope of 3 , not -3 .

Choice B is incorrect. This inequality represents a region below, not above, the boundary line shown.

Choice C is incorrect. This inequality represents a region whose boundary line has a slope of 3 , not -3 .

Question 278 278 of 569 selected Linear Inequalities In 1 Or 2 Variables H

A team hosting an event to raise money for new uniforms plans to sell at least 140 tickets before this event and at least 220 tickets during this event to raise a total of at least $5,820 from all tickets sold. The price of a ticket during this event is $3 less than the price of a ticket before this event. Which inequality represents this situation, where x is the price, in dollars, of a ticket sold during this event?

  1. 140(x+3)+220x5,820

  2. 140(x+3)+220x5,820

  3. 140(x-3)+220x5,820

  4. 140(x-3)+220x5,820

Show Answer Correct Answer: B

Choice B is correct. It’s given that a team plans to sell at least 140 tickets before an event and at least 220 tickets during the event to raise a total of at least $5,820 from all tickets sold. It’s also given that the price of a ticket during the event is $3 less than the price of a ticket before the event and that x represents the price, in dollars, of a ticket sold during the event. It follows that x+3 represents the price, in dollars, of a ticket sold before the event. The expression 140(x+3) represents the planned revenue, in dollars, from the tickets sold before the event, and the expression 220x represents the planned revenue, in dollars, from the tickets sold during the event. Thus, the expression 140(x+3)+220x represents the planned revenue, in dollars, from all tickets sold. Since the team plans to raise a total of at least $5,820 from all tickets sold, the total revenue must be at least $5,820. Therefore, the inequality 140(x+3)+220x5,820 represents this situation.

Choice A is incorrect. This inequality represents a situation in which the team raises a total of at most $5,820 from all tickets sold.

Choice C is incorrect. This inequality represents a situation in which the price of a ticket before the event is $3 less, rather than $3 more, than the price of a ticket during the event and the team raises a total of at most $5,820 from all tickets sold.

Choice D is incorrect. This inequality represents a situation in which the price of a ticket before the event is $3 less, rather than $3 more, than the price of a ticket during the event.

Question 279 279 of 569 selected Linear Equations In 2 Variables E

7 x - 4 y = -84

For the given equation, which table gives three values of x and their corresponding values of y ?

Show Answer Correct Answer: A

Choice A is correct. To verify which table represents this linear relationship, the values in each table can be checked by substituting them into the given equation. The table in choice A shows that when x=0, y=21. Substituting these values into the given equation yields 7(0)4(21)=-84, or 84=-84, which is true. Additionally, the table in choice A shows that when x=4, y=28. Substituting these values into the given equation yields 7(4)4(28)=-84, or 84=-84, which is true. Finally, the table in choice A shows that when x=8y=35. Substituting these values into the given equation yields 7(8)4(35)=-84, or 84=-84, which is true. Therefore, the table in choice A gives three values of x and their corresponding values of y.

Choice B is incorrect. The table in choice B shows that when x=0, y=35. Substituting these values into the given equation yields 7(0)4(35)=-84, or 140=-84, which is not true.

Choice C is incorrect. The table in choice C shows that when x=21, y=0. Substituting these values into the given equation yields 7(21)4(0)=-84, or 147=-84, which is not true.

Choice D is incorrect. The table in choice D shows that when x=21, y=8. Substituting these values into the given equation yields 7(21)4(8)=-84, or 115=-84, which is not true.

Question 280 280 of 569 selected Linear Inequalities In 1 Or 2 Variables H

y<5x+6

For which of the following tables are all the values of x and their corresponding values of y solutions to the given inequality?

  1. x y
    3 17
    5 27
    7 37
  2. x y
    3 17
    5 35
    7 37
  3. x y
    3 25
    5 35
    7 45
  4. x y
    3 21
    5 31
    7 41
Show Answer Correct Answer: A

Choice A is correct. Substituting 3 for x in the given inequality yields y<5(3)+6, or y<21. Therefore, when x = 3 , the corresponding value of y is less than 21 . Substituting 5 for x in the given inequality yields y<5(5)+6, or y<31. Therefore, when x = 5 , the corresponding value of y is less than 31 . Substituting 7 for x in the given inequality yields y<5(7)+6, or y<41. Therefore, when x = 7 , the corresponding value of y is less than 41 . For the table in choice A, when x = 3 , the corresponding value of y is 17 , which is less than 21 ; when x = 5 , the corresponding value of y is 27 , which is less than 31 ; and when x = 7 , the corresponding value of y is 37 , which is less than 41 . Therefore, the table in choice A gives values of x and their corresponding values of y that are all solutions to the given inequality.

Choice B is incorrect and may result from conceptual or calculation errors. 

Choice C is incorrect and may result from conceptual or calculation errors. 

Choice D is incorrect and may result from conceptual or calculation errors. 

Question 281 281 of 569 selected Linear Inequalities In 1 Or 2 Variables E

A geologist needs to collect at least 67 samples of lava from a volcano. If the geologist has already collected 63 samples from the volcano, what is the minimum number of additional samples the geologist needs to collect?

  1. 130

  2. 63

  3. 4

  4. 0

Show Answer Correct Answer: C

Choice C is correct. It's given that the geologist has already collected 63 samples from the volcano. Let x represent the number of additional samples the geologist needs to collect. After collecting x additional samples, the geologist will have collected a total of 63+x samples. It's given that the geologist needs to collect at least 67 samples. Therefore, 63+x67. Subtracting 63 from each side of this inequality yields the inequality x4. Thus, the geologist needs to collect a minimum of 4 additional samples.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect. This is the number of samples the geologist has already collected, rather than the minimum number of additional samples the geologist needs to collect.

Choice D is incorrect. If the geologist collects 0 additional samples, the geologist will have collected a total of 63 samples, which is less than 67 samples.

Question 282 282 of 569 selected Linear Functions E

For the function f , the graph of y=f(x) in the xy-plane has a slope of 3 and passes through the point (0,-8). Which equation defines f ?

  1. f(x)=3x

  2. f(x)=3x-8

  3. f(x)=3x+5

  4. f(x)=3x+11

Show Answer Correct Answer: B

Choice B is correct. An equation defining a linear function can be written in the form f(x)=mx+b, where m and b are constants, m is the slope of the graph of y=f(x) in the xy-plane, and (0,b) is the y-intercept of the graph. It's given that for the function f , the graph of y=f(x) in the xy-plane has a slope of 3 . Therefore, m=3. It's also given that this graph passes through the point (0,-8). Therefore, the y-intercept of the graph is (0,-8), and it follows that b=-8. Substituting 3 for m and -8 for b in the equation f(x)=mx+b yields f(x)=3x-8. Thus, the equation that defines f is f(x)=3x-8.

Choice A is incorrect. For this function, the graph of y=f(x) in the xy-plane passes through the point (0,0), not (0,-8).

Choice C is incorrect. For this function, the graph of y=f(x) in the xy-plane passes through the point (0,5), not (0,-8).

Choice D is incorrect. For this function, the graph of y=f(x) in the xy-plane passes through the point (0,11), not (0,-8).

Question 283 283 of 569 selected Linear Equations In 2 Variables M

A batch of banana milkshakes consists of 4 cups of ice cream and 2 bananas and has 1,114 milligrams (mg) of calcium. There is 276 mg of calcium in 1 cup of the ice cream used to make this batch of milkshakes. How much calcium, in mg, is in 1 banana?

  1. 5

  2. 10

  3. 419

  4. 1,104

Show Answer Correct Answer: A

Choice A is correct. It’s given that a batch of banana milkshakes consists of 4 cups of ice cream and 2 bananas and has 1,114 mg of calcium. It’s also given that there is 276 mg of calcium in 1 cup of the ice cream used to make this batch of milkshakes. It follows that the total amount of calcium in 4 cups of ice cream is 4(276), or 1,104 mg. Let x represent the amount of calcium, in mg, in 1 banana. It follows that the total amount of calcium in 2 bananas is 2x mg. Since the batch of banana milkshakes has 1,114 mg of calcium, the equation 1,104+2x=1,114 represents this situation. Subtracting 1,104 from both sides of this equation yields 2x=10. Dividing both sides of this equation by 2 yields x=5. Therefore, the amount of calcium in 1 banana is 5 mg.

Choice B is incorrect. This is the amount of calcium, in mg, in 2 bananas, not in 1 banana.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect. This is the amount of calcium, in mg, in 4 cups of ice cream, not in 1 banana.

Question 284 284 of 569 selected Linear Equations In 2 Variables E

In the xy-plane, a line has a slope of 6 and passes through the point with coordinates zero comma 8. Which of the following is an equation of this line?

  1. y equals, 6 x plus 8

  2. y equals, 6 x plus 48

  3. y equals, 8 x plus 6

  4. y equals, 8 x plus 48

Show Answer Correct Answer: A

Choice A is correct. The slope-intercept form of an equation for a line is y equals, m x plus b, where m is the slope of the line and b is the y-coordinate of the y-intercept of the line. It’s given that the slope is 6, so m equals 6. It’s also given that the line passes through the point with coordinates 0 comma 8 on the y-axis, so b equals 8 . Substituting m equals 6 and b equals 8 into the equation y equals, m x plus b gives y equals, 6 x plus 8.

Choices B, C, and D are incorrect and may result from misinterpreting the slope-intercept form of an equation of a line.

 

Question 285 285 of 569 selected Linear Inequalities In 1 Or 2 Variables H

The triangle inequality theorem states that the sum of any two sides of a triangle must be greater than the length of the third side. If a triangle has side lengths of 6 and 12 , which inequality represents the possible lengths, x , of the third side of the triangle?

  1. x<18

  2. x>18

  3. 6 <x<18

  4. x<6 or x>18

Show Answer Correct Answer: C

Choice C is correct. It’s given that a triangle has side lengths of 6 and 12 , and x represents the length of the third side of the triangle. It’s also given that the triangle inequality theorem states that the sum of any two sides of a triangle must be greater than the length of the third side. Therefore, the inequalities 6+x>12, 6+12>x, and 12+x>6 represent all possible values of x . Subtracting 6 from both sides of the inequality 6+x>12 yields x>12-6, or x>6. Adding 6 and 12 in the inequality 6+12>x yields 18>x, or x<18. Subtracting 12 from both sides of the inequality 12+x>6 yields x>6-12, or x>-6. Since all x-values that satisfy the inequality x>6 also satisfy the inequality x>-6, it follows that the inequalities x>6 and x<18 represent the possible values of x . Therefore, the inequality 6<x<18 represents the possible lengths, x , of the third side of the triangle.

Choice A is incorrect. This inequality gives the upper bound for x but does not include its lower bound.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 286 286 of 569 selected Systems Of 2 Linear Equations In 2 Variables M

y = 9 x + 12

x + 7 y = 20

The solution to the given system of equations is (x,y). What is the value of y ?

Show Answer Correct Answer: 3

The correct answer is 3 . It’s given that y = 9 x + 12 . Substituting 9 x + 12 for y in the second equation in the system, x + 7 y = 20 , yields x+7(9x+12)=20, which gives x+63x+84=20, or 64 x + 84 = 20 . Subtracting 84 from each side of this equation yields 64 x = - 64 . Dividing each side of this equation by 64 yields x = - 1 . Substituting - 1 for x in the first equation in the system, y = 9 x + 12 , yields y=9(-1)+12, or y = 3 . Therefore, the value of y is 3 .

Question 287 287 of 569 selected Linear Equations In 1 Variable M

If 5(x+4)=4(x+4)+29, what is the value of x+4?

  1. -4

  2. 25

  3. 29

  4. 33

Show Answer Correct Answer: C

Choice C is correct. Subtracting 4(x+4) from both sides of the given equation yields 1(x+4)=29, or x + 4 = 29 . Therefore, the value of x + 4 is 29 .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect. This is the value of x , not x + 4 .

Choice D is incorrect and may result from conceptual or calculation errors.

Question 288 288 of 569 selected Linear Equations In 2 Variables M

An artist paints and sells square tiles. The selling price P, in dollars, of a painted tile is a linear function of the side length of the tile s, in inches, as shown in the table below.

Side length, s (inches)Price, P (dollars)
38.00
618.00
928.00
 

Which of the following could define the relationship between s and P ?

  1. P equals, 3 s plus 10

  2. P equals, ten thirds s, plus 8

  3. P equals, ten thirds s, minus 2

  4. P equals, three tenths s, minus one tenth

Show Answer Correct Answer: C

Choice C is correct. The relationship between s and P can be modeled by a linear equation of the form P = ks + a, where k and a are constants. The table shows that P increases by 10 when s increases by 3, so k = the fraction 10 over 3. To solve for a, substitute one of the given pairs of values for s and P: when s = 3, P = 8, so 8 equals, the fraction 10 over 3, end fraction, times 3, plus a, which yields a = –2. The solution is therefore P equals, the fraction 10 over 3, end fraction, times s minus 2.

Choice A is incorrect. When s = 3, P = 8, but 3(3) + 10 = 19 ≠︀ 8. Choice B is incorrect. This may result from using the first number given for P in the table as the constant term in the linear equation P = ks + a, which is true only when s = 0. Choice D is incorrect and may result from using the reciprocal of the slope of the line.

 

Question 289 289 of 569 selected Linear Functions M

The function f(x)=55.20-0.16x gives the estimated surface water temperature f(x), in degrees Celsius, of a body of water on the x th day of the year, where 220x360. Based on the model, what is the estimated surface water temperature, in degrees Celsius, of this body of water on the 326th day of the year?

  1. 55.20

  2. 3.04

  3. -0.16

  4. -52.16

Show Answer Correct Answer: B

Choice B is correct. It’s given that the function f(x)=55.20-0.16x gives the estimated surface water temperature, in degrees Celsius, of a body of water on the xth day of the year. Substituting 326 for x in the given function yields f(326)=55.20-0.16(326), which is equivalent to f(326)=55.20-52.16, or f(326)=3.04. Therefore, the estimated surface water temperature, in degrees Celsius, of this body of water on the 326th day of the year is 3.04.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect. This is the rate of change, in degrees Celsius per day, of the estimated surface water temperature.

Choice D is incorrect. This is the change, in degrees Celsius, in the estimated surface water temperature over 326 days.

Question 290 290 of 569 selected Linear Equations In 1 Variable E

Cathy has n CDs. Gerry has 3 more than twice the number of CDs that Cathy has. In terms of n, how many CDs does Gerry have?

  1. 3 n minus 2

  2. 3 n plus 2

  3. 2 n minus 3

  4. 2 n plus 3

Show Answer Correct Answer: D

Choice D is correct. The term 2n represents twice the number of CDs that Cathy has, and adding 3 represents 3 more than that amount.

Choices A and B are incorrect. The expression 3n represents three times the number of CDs that Cathy has. Choice C is incorrect. Subtracting 3 represents 3 fewer than twice the number of CDs that Cathy has.

 

Question 291 291 of 569 selected Linear Equations In 2 Variables E

The equation 40 x + 20 y = 160 represents the number of sweaters, x , and number of shirts, y , that Yesenia purchased for $160. If Yesenia purchased 2 sweaters, how many shirts did she purchase?  

  1. 3

  2. 4

  3. 8

  4. 40

Show Answer Correct Answer: B

Choice B is correct. It's given that the equation 40 x + 20 y = 160 represents the number of sweaters, x , and the number of shirts, y , that Yesenia purchased for $160. If Yesenia purchased 2 sweaters, the number of shirts she purchased can be calculated by substituting 2 for x in the given equation, which yields 40(2)+20y=160, or 80+20y=160. Subtracting 80 from both sides of this equation yields 20 y = 80 . Dividing both sides of this equation by 20 yields y = 4 . Therefore, if Yesenia purchased 2 sweaters, she purchased 4 shirts.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect. This is the number of shirts Yesenia purchased if she purchased 0 sweaters.

Choice D is incorrect. This is the price, in dollars, for each sweater, not the number of shirts Yesenia purchased. 

Question 292 292 of 569 selected Linear Functions E

Marisol drove 3 hours from City A to City B. The equation below estimates the distance d, in miles, Marisol traveled after driving for t hours.

d equals 45 t

Which of the following does 45 represent in the equation?

  1. Marisol took 45 trips from City A to City B.

  2. The distance between City A and City B is 45 miles.

  3. Marisol drove at an average speed of about 45 miles per hour.

  4. It took Marisol 45 hours to drive from City A to City B.

Show Answer Correct Answer: C

Choice C is correct. It’s given that d is the distance, in miles, Marisol traveled after driving for t hours. Therefore, 45 represents the distance in miles traveled per hour, which is the speed she drove in miles per hour.

Choice A is incorrect and may result from misidentifying speed as the number of trips. Choice B is incorrect and may result from misidentifying speed as the total distance. Choice D is incorrect and may result from misidentifying the speed as the time, in hours.

 

Question 293 293 of 569 selected Linear Equations In 1 Variable E

If x = 7 , what is the value of x + 20 ?

  1. 13

  2. 20

  3. 27

  4. 34

Show Answer Correct Answer: C

Choice C is correct. It’s given that x = 7 . Substituting 7 for x into the given expression x + 20 yields 7+20, which is equivalent to 27 .

Choice A is incorrect. This is the value of x + 6 .

Choice B is incorrect. This is the value of x + 13 .

Choice D is incorrect. This is the value of x + 27 .

Question 294 294 of 569 selected Linear Functions E

For the linear function f , f(0)=17 and f(1)=17. Which equation defines f ?

  1. f(x)=117

  2. f(x)=1

  3. f(x)=17

  4. f(x)=34

Show Answer Correct Answer: C

Choice C is correct. An equation defining the linear function f can be written in the form f(x)=mx+b, where m is the slope and (0,b) is the y-intercept of the graph of y=f(x) in the xy-plane. The slope of the graph of y=f(x) can be found using the formula m=y2-y1x2-x1, where (x1,y1) and (x2,y2) are any two points that the graph passes through. If f(0)=17, it follows that the graph of y=f(x) passes through the point (0,17). If f(1)=17, it follows that the graph of y=f(x) passes through the point (1,17). Substituting (0,17) and (1,17) for (x1,y1) and (x2,y2), respectively, in the formula m=y2-y1x2-x1 yields m=17-171-0, which is equivalent to m=01, or m=0. Since the graph of y=f(x) passes through (0,17), it follows that b=17. Substituting 0 for m and 17 for b in the equation f(x)=mx+b yields f(x)=0x+17, or f(x)=17. Thus, the equation that defines f is f(x)=17.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 295 295 of 569 selected Linear Equations In 1 Variable E

A gym charges its members a onetime $36 enrollment fee and a membership fee of $19 per month. If there are no charges other than the enrollment fee and the membership fee, after how many months will a member have been charged a total of $188 at the gym?

  1. 4

  2. 5

  3. 8

  4. 10

Show Answer Correct Answer: C

Choice C is correct. It’s given that a gym charges its members a onetime $36 enrollment fee and a membership fee of $19 per month. Let x represent the number of months at the gym after which a member will have been charged a total of $188. If there are no charges other than the enrollment fee and the membership fee, the equation 36+19x=188 can be used to represent this situation. Subtracting 36 from both sides of this equation yields 19x=152. Dividing both sides of this equation by 19 yields x=8. Therefore, a member will have been charged a total of $188 after 8 months at the gym.

Choice A is incorrect. A member will have been charged a total of $(36+19×4), or $112, after 4 months at the gym.

Choice B is incorrect. A member will have been charged a total of $(36+19×5), or $131, after 5 months at the gym.

Choice D is incorrect. A member will have been charged a total of $(36+19×10), or $226, after 10 months at the gym.

Question 296 296 of 569 selected Linear Equations In 1 Variable E

If 7 x = 28 , what is the value of 8 x ?

  1. 21

  2. 32

  3. 168

  4. 224

Show Answer Correct Answer: B

Choice B is correct. Dividing both sides of the given equation 7 x = 28 by 7 yields x = 4 . Substituting 4 for x in the expression 8 x yields 8(4), which is equivalent to 32 .

Choice A is incorrect. This is the value of 214x.

Choice C is incorrect. This is the value of 42 x .

Choice D is incorrect. This is the value of 56 x .

Question 297 297 of 569 selected Systems Of 2 Linear Equations In 2 Variables M

x + 2 y = 6

x - 2 y = 4

The solution to the given system of equations is (x,y). What is the value of x ?

  1. 2.5

  2. 5

  3. 6

  4. 10

Show Answer Correct Answer: B

Choice B is correct. Adding the first equation to the second equation in the given system yields (x+2y)+(x-2y)=6+4, or (x+x)+(2y-2y)=10. Combining like terms in this equation yields 2x=10. Dividing both sides of this equation by 2 yields x = 5 . Thus, the value of x is 5 .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect. This is the value of 2 x , not x .

Question 298 298 of 569 selected Linear Equations In 1 Variable H
 
Townsend Realty Group Investments
Property addressPurchase price (dollars)Monthly rental price (dollars)
Clearwater Lane128,000950
Driftwood Drive176,0001,310
Edgemont Street70,000515
Glenview Street140,0001,040
Hamilton Circle450,0003,365

The Townsend Realty Group invested in the five different properties listed in the table above. The table shows the amount, in dollars, the company paid for each property and the corresponding monthly rental price, in dollars, the company charges for the property at each of the five locations. Townsend Realty purchased the Glenview Street property and received a 40% discount off the original price along with an additional 20% off the discounted price for purchasing the property in cash. Which of the following best approximates the original price, in dollars, of the Glenview Street property?

  1. $350,000

  2. $291,700

  3. $233,300

  4. $175,000

Show Answer Correct Answer: B

Choice B is correct. Let x be the original price, in dollars, of the Glenview Street property. After the 40% discount, the price of the property became 0 point 6 x dollars, and after the additional 20% off the discounted price, the price of the property became 0 point 8 times 0 point 6 x. Thus, in terms of the original price of the property, x, the purchase price of the property is 0 point 4 8 x . It follows that 0 point 4 8 x equals 140,000. Solving this equation for x gives x equals 291,666 point 6, with the 6 repeating after the decimal point. Therefore, of the given choices, $291,700 best approximates the original price of the Glenview Street property.

Choice A is incorrect because it is the result of dividing the purchase price of the property by 0.4, as though the purchase price were 40% of the original price. Choice C is incorrect because it is the closest to dividing the purchase price of the property by 0.6, as though the purchase price were 60% of the original price. Choice D is incorrect because it is the result of dividing the purchase price of the property by 0.8, as though the purchase price were 80% of the original price.

Question 299 299 of 569 selected Linear Equations In 1 Variable E

If 4 x + 2 = 12 , what is the value of 16 x + 8 ?

  1. 40

  2. 48

  3. 56

  4. 60

Show Answer Correct Answer: B

Choice B is correct. Multiplying both sides of the given equation by 4 yields (4)(4x+2)=(4)(12), or 16x+8=48. Therefore, the value of 16x+8 is 48 .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 300 300 of 569 selected Linear Functions M

A linear model estimates the population of a city from 1991 to 2015 . The model estimates the population was 57 thousand in 1991 , 224 thousand in 2011 , and x thousand in 2015 . To the nearest whole number, what is the value of x ?

Show Answer Correct Answer: 257

The correct answer is 257. It’s given that a linear model estimates the population of a city from 1991 to 2015. Since the population can be estimated using a linear model, it follows that there is a constant rate of change for the model. It’s also given that the model estimates the population was 57 thousand in 1991, 224 thousand in 2011, and x thousand in 2015. The change in the population between 2011 and 1991 is 224-57, or 167, thousand. The change in the number of years between 2011 and 1991 is 2011-1991, or 20, years. Dividing 167 by 20 gives 167/20, or 8.35, thousand per year. Thus, the change in population per year from 1991 to 2015 estimated by the model is 8.35 thousand. The change in the number of years between 2015 and 2011 is 2015-2011, or 4, years. Multiplying the change in population per year by the change in number of years yields the increase in population from 2011 to 2015 estimated by the model: (8.35)(4), or 33.4, thousand. Adding the change in population from 2011 to 2015 estimated by the model to the estimated population in 2011 yields the estimated population in 2015. Thus, the estimated population in 2015 is 33.4+224, or 257.4, thousand. Therefore to the nearest whole number, the value of x is 257.

Question 301 301 of 569 selected Linear Inequalities In 1 Or 2 Variables M

y>4x+8

For which of the following tables are all the values of x and their corresponding values of y solutions to the given inequality?

Show Answer Correct Answer: A

Choice A is correct. In each choice, the values of x are 2 , 4 , and 6 . Substituting the first value of x , 2 , for x in the given inequality yields y>4(2)+8, or y>16. Therefore, when x = 2 , the corresponding value of y must be greater than 16 . Of the given choices, only choice A is a table where the value of y corresponding to x = 2 is greater than 16 . To confirm that the other values of x in this table and their corresponding values of y are also solutions to the given inequality, the values of x and y in the table can be substituted for x and y in the given inequality. Substituting 4 for x and 30 for y in the given inequality yields 30>4(4)+8, or 30>24, which is true. Substituting 6 for x and 41 for y in the given inequality yields 41>4(6)+8, or 41>32, which is true. It follows that for choice A, all the values of x and their corresponding values of y are solutions to the given inequality.

Choice B is incorrect. Substituting 2 for x and 8 for y in the given inequality yields 8>4(2)+8, or 8>16, which is false.

Choice C is incorrect. Substituting 2 for x and 13 for y in the given inequality yields 13>4(2)+8, or 13>16, which is false.

Choice D is incorrect. Substituting 2 for x and 13 for y in the given inequality yields 13>4(2)+8, or 13>16, which is false.

Question 302 302 of 569 selected Linear Equations In 2 Variables E
x y
0 18
1 13
2 8

The table shows three values of x and their corresponding values of y . There is a linear relationship between x and y . Which of the following equations represents this relationship?

  1. y = 18 x + 13

  2. y = 18 x + 18

  3. y = - 5 x + 13

  4. y = - 5 x + 18

Show Answer Correct Answer: D

Choice D is correct. A linear relationship can be represented by an equation of the form y = m x + b , where m and b are constants. It’s given in the table that when x = 0 , y = 18 . Substituting 0 for x and 18 for y in y = m x + b yields 18=m(0)+b, or 18 = b . Substituting 18 for b in the equation y = m x + b yields y = m x + 18 . It’s also given in the table that when x = 1 , y = 13 . Substituting 1 for x and 13 for y in the equation y = m x + 18 yields 13=m(1)+18, or 13 = m + 18 . Subtracting 18 from both sides of this equation yields -5 = m . Therefore, the equation y = - 5 x + 18 represents the relationship between x and y .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Question 303 303 of 569 selected Systems Of 2 Linear Equations In 2 Variables H

6+7r=pw

7 r - 5 w = 5 w + 11

In the given system of equations, p is a constant. If the system has no solution, what is the value of p ?

Show Answer Correct Answer: 10

The correct answer is 10 . Solving by substitution, the given system of equations, where p is a constant, can be written so that the left-hand side of each equation is equal to 7 r . Subtracting 6 from each side of the first equation in the given system, 6+7r=pw, yields 7r=pw-6. Adding 5 w to each side of the second equation in the given system, 7r-5w=5w+11, yields 7r=10w+11. Since the left-hand side of each equation is equal to 7 r , setting the the right-hand side of the equations equal to each other yields pw-6=10w+11. A linear equation in one variable, w , has no solution if and only if the equation is false; that is, when there's no value of w that produces a true statement. For the equation pw-6=10w+11, there's no value of w that produces a true statement when p w = 10 w . Therefore, for the equation p w - 6 = 10 w + 11 , there's no value of w that produces a true statement when the value of p is 10 . It follows that in the given system of equations, the system has no solution when the value of p is 10 .

Question 304 304 of 569 selected Linear Equations In 1 Variable E

On the first day of a semester, a film club has 90 members. Each day after the first day of the semester, 10 new members join the film club. If no members leave the film club, how many total members will the film club have 4 days after the first day of the semester?

  1. 400

  2. 130

  3. 94

  4. 90

Show Answer Correct Answer: B

Choice B is correct. It’s given that the film club has 90 members on the first day of a semester, and 10 new members join the film club each day after the first day of the semester. This means that after 4 days, 4×10, or 40 , new members will have joined the club. Adding 40 members to the original 90 club members yields 130 members. Thus, the film club will have 130 total members 4 days after the first day of the semester.

Choice A is incorrect. This is the number of members that will have joined the film club 4 days after the first day of the semester if 100 new members, not 10 , join the film club each day.

Choice C is incorrect. This is the number of members the film club will have 4 days after the first day of the semester if 1 new member, not 10 , joins the film club each day.

Choice D is incorrect. This is the number of members the film club has on the first day of the semester.

Question 305 305 of 569 selected Linear Equations In 1 Variable E

For what value of w does w minus 10, equals, 2 times, open parenthesis, w plus 5, close parenthesis ?

  1. 0

  2. negative 15

  3. negative 20

Show Answer Correct Answer: D

Choice D is correct. To solve the equation, use the distributive property to multiply on the right-hand side of the equation which gives w 10 = 2w + 10. Subtract w from both sides of the equation, which gives –10 = w + 10. Finally, subtract 10 from both sides of the equation, which gives –20 = w.

Choices A and B are incorrect and may result from making sign errors. Choice C is incorrect and may result from incompletely distributing the 2 in the expression 2(w + 5).

Question 306 306 of 569 selected Linear Inequalities In 1 Or 2 Variables E

  • The boundary of the inequality is a solid line.
    • The line slants sharply down from left to right.
    • The line passes through the following points:
      • (1.0 comma 6.5)
      • (3.0 comma negative 4.5)
  • The area above and to the right of the boundary is shaded.

The shaded region shown represents solutions to an inequality. Which ordered pair (x,y) is a solution to this inequality?

  1. (0,-4)

  2. (0,4)

  3. (-4,0)

  4. (4,0)

Show Answer Correct Answer: D

Choice D is correct. Since the shaded region shown represents solutions to an inequality, an ordered pair (x,y) is a solution to the inequality if it's represented by a point in the shaded region. Of the given choices, only (4,0) is represented by a point in the shaded region. Therefore, (4,0) is a solution to the inequality.

Choice A is incorrect and may result from conceptual errors.

Choice B is incorrect and may result from conceptual errors.

Choice C is incorrect and may result from conceptual errors.

Question 307 307 of 569 selected Linear Functions M

In the linear function f , f(0)=8 and f(1)=12. Which equation defines f ?

  1. f(x)=12x+8

  2. f(x)=4x

  3. f(x)=4x+12

  4. f(x)=4x+8

Show Answer Correct Answer: D

Choice D is correct. Since f is a linear function, it can be defined by an equation of the form f(x)=ax+b, where a and b are constants. It's given that f(0)=8. Substituting 0 for x and 8 for f(x) in the equation f(x)=ax+b yields 8=a(0)+b, or 8=b. Substituting 8 for b in the equation f(x)=ax+b yields f(x)=ax+8. It's given that f(1)=12. Substituting 1 for x and 12 for f(x) in the equation f(x)=ax+8 yields 12=a(1)+8, or 12=a+8. Subtracting 8 from both sides of this equation yields a=4. Substituting 4 for a in the equation f(x)=ax+8 yields f(x)=4x+8. Therefore, an equation that defines f is f(x)=4x+8.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Question 308 308 of 569 selected Linear Functions M

f(x)=2x+244

The given function f represents the perimeter, in centimeters (cm), of a rectangle with a length of x cm and a fixed width. What is the width, in cm, of the rectangle?

  1. 2

  2. 122

  3. 244

  4. 488

Show Answer Correct Answer: B

Choice B is correct. It's given that f(x)=2x+244 represents the perimeter, in centimeters (cm), of a rectangle with a length of x  cm and a fixed width. If w represents a fixed width, in cm, then the perimeter, in cm, of a rectangle with a length of x cm and a fixed width of w cm can be given by the function f(x)=2x+2w. Therefore, 2x+2w=2x+244. Subtracting 2 x from both sides of this equation yields 2w=244. Dividing both sides of this equation by 2 yields w=122. Therefore, the width, in cm, of the rectangle is 122 .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 309 309 of 569 selected Systems Of 2 Linear Equations In 2 Variables M

- 15 x + 25 y = 65

One of the two equations in a system of linear equations is given. The system has infinitely many solutions. Which of the following could be the second equation in the system?

  1. 12 x + 20 y = 52

  2. 12 x + 20 y = -52

  3. - 12 x + 20 y = 52

  4. - 12 x + 20 y = -52

Show Answer Correct Answer: C

Choice C is correct. It's given that the system has infinitely many solutions. A system of two linear equations has infinitely many solutions when the two linear equations are equivalent. Dividing both sides of the given equation by 5 yields - 3 x + 5 y = 13 . Dividing both sides of choice C by 4 also yields - 3 x + 5 y = 13 , so choice C is equivalent to the given equation. Thus, choice C could be the second equation in the system.

Choice A is incorrect. The system consisting of this equation and the given equation has one solution, not infinitely many solutions.

Choice B is incorrect. The system consisting of this equation and the given equation has one solution, not infinitely many solutions.

Choice D is incorrect. The system consisting of this equation and the given equation has no solution, not infinitely many solutions.

Question 310 310 of 569 selected Linear Functions M

The function f is defined by f(x)=-9x+9. What is the y -coordinate of the y -intercept of the graph of y=f(x) in the x y -plane?

Show Answer Correct Answer: 9

The correct answer is 9. The y-intercept of the graph of y=f(x) in the xy-plane is the point where the graph of y=f(x) crosses the y-axis, which occurs at x=0. It’s given that the function f is defined by f(x)=-9x+9. Substituting 0 for x and y for f(x) in this equation yields y=-9(0)+9, or y=9. It follows that y=9 when x=0 and that the y-intercept of the graph of y=f(x) in the xy-plane is (0,9). Therefore, the y-coordinate of the y-intercept of the graph of y=f(x) in the xy-plane is 9.

Question 311 311 of 569 selected Linear Equations In 2 Variables H

Line l in the xy-plane is perpendicular to the line with equation x equals 2. What is the slope of line l?

  1. 0

  2. negative one half

  3. negative 2

  4. The slope of line l is undefined.

Show Answer Correct Answer: A

Choice A is correct. It is given that line l is perpendicular to a line whose equation is x = 2. A line whose equation is a constant value of x is vertical, so l must therefore be horizontal. Horizontal lines have a slope of 0, so l has a slope of 0.

Choice B is incorrect. A line with slope negative one half is perpendicular to a line with slope 2. However, the line with equation x = 2 is vertical and has undefined slope (not slope of 2). Choice C is incorrect. A line with slope –2 is perpendicular to a line with slope one half. However, the line with equation x = 2 has undefined slope (not slope of one half). Choice D is incorrect; this is the slope of the line x = 2 itself, not the slope of a line perpendicular to it.

 

Question 312 312 of 569 selected Systems Of 2 Linear Equations In 2 Variables E

x = 5

y = x - 8

Which of the following points (x,y) is the solution to the given system of equations in the xy-plane?

  1. (0,0)

  2. (5,-3)

  3. (5,-8)

  4. (5,8)

Show Answer Correct Answer: B

Choice B is correct. A solution to a system of equations in the xy-plane is a point (x,y) that lies on the graph of each equation in the system. The first equation given is x = 5 . Substituting 5 for x in the second given equation yields y=5-8, or y = -3 . It follows that in the xy-plane, the point (5,-3) lies on the graph of each equation in the system. Therefore, the solution to the given system of equations in the xy-plane is (5,-3).

Choice A is incorrect. The point (0,0) doesn't lie on the graph of either equation in the given system.

Choice C is incorrect. The point (5,-8) doesn't lie on the graph of the second equation in the given system.

Choice D is incorrect. The point (5,8) doesn't lie on the graph of the second equation in the given system.

Question 313 313 of 569 selected Linear Functions M

w(t)=300-4t

The function w models the volume of liquid, in milliliters, in a container t seconds after it begins draining from a hole at the bottom. According to the model, what is the predicted volume, in milliliters, draining from the container each second?

  1. 300

  2. 296

  3. 75

  4. 4

Show Answer Correct Answer: D

Choice D is correct. It’s given that the function w models the volume of liquid, in milliliters, in a container t seconds after it begins draining from a hole at the bottom. The given function w(t)=300-4t can be rewritten as w(t)=-4t+300. Thus, for each increase of t by 1 , the value of w(t) decreases by 4(1), or 4 . Therefore, the predicted volume, in milliliters, draining from the container each second is 4 milliliters.

Choice A is incorrect. This is the amount of liquid, in milliliters, in the container before the liquid begins draining.

Choice B is incorrect and may result from conceptual errors.

Choice C is incorrect and may result from conceptual errors.

Question 314 314 of 569 selected Linear Equations In 2 Variables M

Line k is defined by y = 7 x + 1 8 . Line j is perpendicular to line k in the xy-plane. What is the slope of line j ?

  1. -8

  2. - 1 7

  3. 1 8

  4. 7

Show Answer Correct Answer: B

Choice B is correct. It's given that line k is defined by y=7x+18. For an equation in slope-intercept form y=mx+b, m represents the slope of the line defined by this equation in the xy-plane and b represents the y-coordinate of the y-intercept of this line. Therefore, the slope of line k is 7 . It’s also given that line j is perpendicular to line k in the xy-plane. Therefore, the slope of line j is the opposite reciprocal of the slope of line k . The opposite reciprocal of 7 is - 1 7 . Therefore, the slope of line j is - 1 7 .

Choice A is incorrect. This is the opposite reciprocal of the y-coordinate of the y-intercept, not the slope, of line k .

Choice C is incorrect. This is the y-coordinate of the y-intercept of line k , not the slope of line j .

Choice D is incorrect. This is the slope of a line that is parallel, not perpendicular, to line k .

Question 315 315 of 569 selected Linear Equations In 1 Variable E

16 x + 30 = 190

Which equation has the same solution as the given equation?

  1. 16 x = 30

  2. 16 x = 130

  3. 16 x = 160

  4. 16 x = 190

Show Answer Correct Answer: C

Choice C is correct. It’s given that 16x+30=190. Subtracting 30 from each side of this equation yields 16x=160. Therefore, the equation 16x=160 is equivalent to the given equation and has the same solution.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 316 316 of 569 selected Linear Equations In 1 Variable M

2 times, open parenthesis, p plus 1, close parenthesis, plus, 8 times, open parenthesis, p minus 1, close parenthesis, equals, 5 p

What value of p is the solution of the equation above?

Show Answer

The correct answer is 1.2. One way to solve the equation 2 times, open parenthesis, p plus 1, close parenthesis, plus, 8 times, open parenthesis, p minus 1, close parenthesis, equals 5 p is to first distribute the terms outside the parentheses to the terms inside the parentheses: 2 p plus 2, plus 8 p, minus 8, equals 5 p. Next, combine like terms on the left side of the equal sign: 10 p minus 6, equals 5 p . Subtracting 10p from both sides yields negative 6 equals negative 5 p . Finally, dividing both sides by negative 5 gives p equals six fifths, which is equivalent to p equals 1 point 2. Note that 1.2 and 6/5 are examples of ways to enter a correct answer.

Question 317 317 of 569 selected Linear Equations In 1 Variable E

If 6+x=9, what is the value of 18+3x?

Show Answer Correct Answer: 27

The correct answer is 27 . Multiplying both sides of the given equation by 3 yields 3(6+x)=3(9), or 18+3x=27. Therefore, the value of 18+3x is 27 .

Question 318 318 of 569 selected Linear Inequalities In 1 Or 2 Variables M

y>14

4x+y<18

The point (x,53) is a solution to the system of inequalities in the xy-plane. Which of the following could be the value of x ?

  1. -9

  2. -5

  3. 5

  4. 9

Show Answer Correct Answer: A

Choice A is correct. It’s given that the point (x,53) is a solution to the given system of inequalities in the xy-plane. This means that the coordinates of the point, when substituted for the variables x and y , make both of the inequalities in the system true. Substituting 53 for y in the inequality y>14 yields 53>14, which is true. Substituting 53 for y in the inequality 4x+y<18 yields 4x+53<18. Subtracting 53 from both sides of this inequality yields 4x<-35. Dividing both sides of this inequality by 4 yields x<-8.75. Therefore, x must be a value less than -8.75 . Of the given choices, only -9 is less than -8.75 .

Choice B is incorrect. Substituting -5 for x and 53 for y in the inequality 4x+y<18 yields 4(-5)+53<18, or 33<18, which is not true.

Choice C is incorrect. Substituting 5 for x and 53 for y in the inequality 4x+y<18 yields 4(5)+53<18, or 73<18, which is not true.

Choice D is incorrect. Substituting 9 for x and 53 for y in the inequality 4x+y<18 yields 4(9)+53<18, or 89<18, which is not true.

Question 319 319 of 569 selected Linear Equations In 1 Variable E

The perimeter of an isosceles triangle is 36 feet. Each of the two congruent sides of the triangle has a length of 10 feet. What is the length, in feet, of the third side?

  1. 10

  2. 12

  3. 16

  4. 18

Show Answer Correct Answer: C

Choice C is correct. It’s given that the perimeter of an isosceles triangle is 36 feet and that each of the two congruent sides has a length of 10 feet. The perimeter of a triangle is the sum of the lengths of its three sides. The equation 10+10+x=36 can be used to represent this situation, where x is the length, in feet, of the third side. Combining like terms on the left-hand side of this equation yields 20+x=36. Subtracting 20 from each side of this equation yields x = 16 . Therefore, the length, in feet, of the third side is 16 .

Choice A is incorrect. This would be the length, in feet, of the third side if the perimeter was 30 feet, not 36 feet.

Choice B is incorrect. This would be the length, in feet, of the third side if the perimeter was 32 feet, not 36 feet.

Choice D is incorrect. This would be the length, in feet, of the third side if the perimeter was 38 feet, not 36 feet.

Question 320 320 of 569 selected Systems Of 2 Linear Equations In 2 Variables E

3x=12

-3x+y=-6

The solution to the given system of equations is (x,y). What is the value of y ?

  1. -3

  2. 6

  3. 18

  4. 30

Show Answer Correct Answer: B

Choice B is correct. Adding the second equation in the given system to the first equation in the given system yields 3x+(-3x+y)=12+(-6), which is equivalent to y=6

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 321 321 of 569 selected Linear Functions H

For the function f , f(cx)=x-8 for all values of x , where c is a positive constant. If f(2)=35, what is the value of c ?

Show Answer Correct Answer: .0465, 2/43

The correct answer is 243. It's given that f(cx)=x-8 for all values of x, where c is a positive constant, and f(2)=35. Therefore, for the given function fcx=2. Dividing both sides of this equation by c yields x=2c. Substituting 2c for x in the equation f(cx)=x-8 yields f(2cc)=2c-8, or f(2)=2c-8. Since it’s given that f(2)=35, substituting 35 for f(2) yields 35=2c-8. Adding 8 to both sides of this equation yields 43=2c. Multiplying both sides of this equation by c yields 43c=2. Dividing both sides of this equation by 43 yields c=243. Note that 2/43, .0465, 0.046, and 0.047 are examples of ways to enter a correct answer.

Question 322 322 of 569 selected Linear Functions H

F(x)=95(x-273.15)+32

The function F gives the temperature, in degrees Fahrenheit, that corresponds to a temperature of x kelvins. If a temperature increased by 2.10 kelvins, by how much did the temperature increase, in degrees Fahrenheit?

  1. 3.78

  2. 35.78

  3. 487.89

  4. 519.89

Show Answer Correct Answer: A

Choice A is correct. It’s given that the function F(x)=95(x-273.15)+32 gives the temperature, in degrees Fahrenheit, that corresponds to a temperature of x kelvins. A temperature that increased by 2.10 kelvins means that the value of x increased by 2.10 kelvins. It follows that an increase in x by 2.10 increases F(x) by 95(2.10), or 3.78. Therefore, if a temperature increased by 2.10 kelvins, the temperature increased by 3.78 degrees Fahrenheit.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 323 323 of 569 selected Linear Functions E

The function f defined by f(t)=14t+9 gives the estimated length, in inches, of a vine plant t months after Tavon purchased it. Which of the following is the best interpretation of 9 in this context?

  1. Tavon will keep the vine plant for 9 months.

  2. The vine plant is expected to grow 9 inches each month.

  3. The vine plant is expected to grow to a maximum length of 9 inches.

  4. The estimated length of the vine plant was 9 inches when Tavon purchased it.

Show Answer Correct Answer: D

Choice D is correct. It's given that the function f defined by f(t)=14t+9 gives the estimated length, in inches, of a vine plant t months after Tavon purchased it. For a function defined by an equation of the form f(t)=mt+b, where m and b are constants, b represents the value of f(0), or the value of f(t) when the value of t is 0 . Therefore, for the function defined by f(t)=14t+9, 9 represents the value of f(t) when the value of t is 0 . This means that 0 months after the vine plant was purchased, the estimated length of the vine plant was 9 inches. Therefore, the best interpretation of 9 in this context is the estimated length of the vine plant was 9 inches when Tavon purchased it.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect. The vine plant is expected to grow 14 inches, not 9 inches, each month.

Choice C is incorrect and may result from conceptual or calculation errors.

Question 324 324 of 569 selected Linear Functions E

In science class, Diego conducted an experiment to learn about evaporation. Diego measured the height of fluid in a beaker over a period of time. The function f(x)=39-0.18x gives the estimated height, in centimeters (cm), of the fluid in the beaker x days after the start of the experiment. Which of the following is the best interpretation of 39 in this context?

  1. The estimated height, in cm, of the fluid at the start of the experiment

  2. The estimated height, in cm, of the fluid at the end of the experiment

  3. The estimated change in the height, in cm, of the fluid each day

  4. The estimated number of days for all the fluid to evaporate

Show Answer Correct Answer: A

Choice A is correct. It’s given that the function f(x)=390.18x gives the estimated height, in centimeters (cm), of the fluid in the beaker x days after the start of the experiment. For a function defined by an equation of the form f(x)=b+mx, where m and b are constants, b represents the value of f(x) when x=0. It follows that in the given function, 39 represents the value of f(x) when x=0. Since x=0 represents the start of the experiment, then the best interpretation of 39 in this context is that the estimated height, in cm, of the fluid was 39 at the start of the experiment.

Choice B is incorrect and may result from conceptual errors.

Choice C is incorrect. The estimated change in the height, in cm, of the fluid each day is 0.18, not 39.

Choice D is incorrect and may result from conceptual errors.

Question 325 325 of 569 selected Linear Functions E

s=40+3t

The equation gives the speed s , in miles per hour, of a certain car t seconds after it began to accelerate. What is the speed, in miles per hour, of the car 5 seconds after it began to accelerate?

  1. 40

  2. 43

  3. 45

  4. 55

Show Answer Correct Answer: D

Choice D is correct. In the given equation, s is the speed, in miles per hour, of a certain car t seconds after it began to accelerate. Therefore, the speed of the car, in miles per hour, 5 seconds after it began to accelerate can be found by substituting 5 for t in the given equation, which yields s=40+3(5), or s=55. Thus, the speed of the car 5 seconds after it began to accelerate is 55 miles per hour.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Question 326 326 of 569 selected Linear Equations In 1 Variable H

Each side of a 30 -sided polygon has one of three lengths. The number of sides with length 8 centimeters (cm) is 5 times the number of sides n with length 3 cm. There are 6 sides with length 4 cm. Which equation must be true for the value of n ?

  1. 5 n + 6 = 30

  2. 6 n + 6 = 30

  3. 8n+3n+4n=30

  4. 8(5n)+3n+4(6)=30

Show Answer Correct Answer: B

Choice B is correct. It’s given that each side of a 30 -sided polygon has one of three lengths. It's also given that the number of sides with length 8 centimeters (cm) is 5 times the number of sides n with length 3 cm. Therefore, there are 5×n, or 5 n , sides with length 8 cm. It’s also given that there are 6 sides with length 4 cm. Therefore, the number of 3 cm, 4 cm, and 8 cm sides are n , 6 , and 5 n , respectively. Since there are a total of 30 sides, the equation n+6+5n=30 represents this situation. Combining like terms on the left-hand side of this equation yields 6n+6=30. Therefore, the equation that must be true for the value of n is 6n+6=30.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 327 327 of 569 selected Systems Of 2 Linear Equations In 2 Variables E

Connor has c dollars and Maria has m dollars. Connor has 4  times as many dollars as Maria, and together they have a total of $25.00 . Which system of equations represents this situation?

  1. c=4m

    c+m=25

  2. m=4c

    c+m=25

  3. c=25m

    c+m=4

  4. m=25c

    c+m=4

Show Answer Correct Answer: A

Choice A is correct. It’s given that Connor has c dollars, Maria has m dollars, and Connor has 4 times as many dollars as Maria. This can be represented by the equation c = 4 m . It’s also given that together, Connor and Maria have a total of $25.00, which can be represented by the equation c+m=25. Therefore, the system consisting of the equations c = 4 m and c+m=25 represents this situation.

Choice B is incorrect. The equation m = 4 c represents a situation where Maria has 4 times as many dollars as Connor, rather than the situation where Connor has 4 times as many dollars as Maria.

Choice C is incorrect. The equation c = 25 m represents a situation where Connor has 25 times, rather than 4 times, as many dollars as Maria. The equation c+m=4 represents a situation where Connor and Maria together have a total of $4.00, rather than $25.00.

Choice D is incorrect. The equation m = 25 c represents a situation where Maria has 25 times as many dollars as Connor, rather than the situation where Connor has 4 times as many dollars as Maria. The equation c+m=4 represents a situation where Connor and Maria together have a total of $4.00, rather than $25.00.

Question 328 328 of 569 selected Linear Functions E
t equals, 1000 plus 18 h

In the equation above, T represents Brittany’s total take-home pay, in dollars, for her first week of work, where h represents the number of hours she worked that week and 1,000 represents a sign-on bonus. If Brittany’s total take-home pay was $1,576, for how many hours was Brittany paid for her first week of work?

  1. 16

  2. 32

  3. 55

  4. 88

Show Answer Correct Answer: B

Choice B is correct. Since Brittany’s total take-home pay was $1,576, the value 1,576 can be substituted for T in the given equation T equals, 1,000 plus 18 h to give 1,576 equals, 1,000 plus 18 h. Subtracting 1,000 from both sides of this equation gives 576 equals 18 h. Dividing both sides of this equation by 18 gives 32 equals h. Therefore, Brittany was paid for 32 hours for her first week of work.

Choice A is incorrect. This is half the number of hours Brittany was paid for. Choice C is incorrect and may result from dividing 1,000 by 18. Choice D is incorrect and may result from dividing 1,576 by 18.

 

Question 329 329 of 569 selected Linear Functions E

The function  is defined as g of x equals, 5 x plus a, where a is a constant. If g of 4 equals 31, what is the value of a ?

  1. negative 23

Show Answer Correct Answer: C

Choice C is correct. Substituting 4 for x in g(x) = 5x + a gives g(4) = 5(4) + a. Since g(4) = 31, the equation g(4) = 5(4) + a simplifies to 31 = 20 + a. It follows that a = 11.

Choices A, B, and D are incorrect and may result from arithmetic errors.

 

Question 330 330 of 569 selected Linear Functions M

j(x)=mx+144

For the linear function j , m is a constant and j(12)=18. What is the value of j(10)?

Show Answer Correct Answer: 39

The correct answer is 39 . It’s given that for the linear function j , m is a constant and j(12)=18. Substituting 12 for x and 18 for j(x) in the given equation yields 18=m(12)+144. Subtracting 144 from both sides of this equation yields -126=m(12). Dividing both sides of this equation by 12 yields -10.5=m. Substituting -10.5 for m in the given equation, j(x)=mx+144, yields j(x)=-10.5x+144. Substituting 10 for x in this equation yields j(10)=(-10.5)(10)+144, or j(10)=39. Therefore, the value of j(10) is 39 .

Question 331 331 of 569 selected Linear Equations In 2 Variables E

Tony spends $80 per month on public transportation. A 10-ride pass costs $12.50, and a single-ride pass costs $1.50. If g represents the number of 10-ride passes Tony buys in a month and t represents the number of single-ride passes Tony buys in a month, which of the following equations best represents the relationship between g and t ?

  1. g plus t, equals 80

  2. g plus t, equals, 1 point 5 0 plus 12 point 5 0

  3. 1 point 5 0 g plus 12 point 5 0 t, equals 80

  4. 12 point 5 0 g plus 1 point 5 0 t, equals 80

Show Answer Correct Answer: D

Choice D is correct. Since a 10-ride pass costs $12.50 and g is the number of 10-ride passes Tony buys in a month, the expression 12 point 5 0 g represents the amount Tony spends on 10-ride passes in a month. Since a single-ride pass costs $1.50 and t is the number of single-ride passes Tony buys in a month, the expression 1 point 5 0 t represents the amount Tony spends on single-ride passes in a month. Therefore, the sum 12 point 5 0 g, plus 1 point 5 0 t represents the amount he spends on the two types of passes in a month. Since Tony spends a total of $80 on passes in a month, this expression can be set equal to 80, producing 12 point 5 0 g, plus 1 point 5 0 t, equals 80.

Choices A and B are incorrect. The expression g plus t represents the total number of the two types of passes Tony buys in a month, not the amount Tony spends, which is equal to 80, nor the cost of one of each pass, which is equal to 1 point 5 0, plus 12 point 5 0. Choice C is incorrect and may result from reversing the cost for each type of pass Tony buys in a month.

 

Question 332 332 of 569 selected Linear Equations In 1 Variable E

If 4 x = 3 , what is the value of 24 x ?

  1. 9 2

  2. 6

  3. 18

  4. 72

Show Answer Correct Answer: C

Choice C is correct. It’s given that 4x=3. Multiplying each side of this equation by 6 yields 24x=18. Therefore, the value of 24 x is 18 .

Choice A is incorrect. This is the value of 6 x , not 24 x .

Choice B is incorrect. This is the value of 8 x , not 24 x .

Choice D is incorrect. This is the value of 96 x , not 24 x .

Question 333 333 of 569 selected Linear Equations In 2 Variables M

In an article about exercise, it is estimated that a 160-pound adult uses 200 calories for every 30 minutes of hiking and 150 calories for every 30 minutes of bicycling. An adult who weighs 160 pounds has completed 1 hour of bicycling. Based on the article, how many hours should the adult hike to use a total of 1,900 calories from bicycling and hiking?

  1. 9.5

  2. 8.75

  3. 6

  4. 4

Show Answer Correct Answer: D

Choice D is correct. Since a 160-pound adult uses 200 calories for every 30 minutes of hiking, then the same adult uses 200 h calories after hiking for h 30-minute periods. Similarly, the same adult uses 150 b calories after bicycling for b 30-minute periods. Therefore, the equation 200 h plus 150 b, equals 1,900 represents the situation where a 160-pound adult uses a total of 1,900 calories from hiking for h 30-minute periods and bicycling for b 30-minute periods. It’s given that the adult completes 1 hour, or 2 30-minute periods, of bicycling. Substituting 2 for b in the equation 200 h plus 150 b, equals 1,900 yields 200 h plus, 150 times 2, equals 1,900, or 200 h plus 300, equals 1,900. Subtracting 300 from both sides of this equation yields 200 h equals 1,600. Dividing both sides by 200 yields h equals 8. Since h represents the number of 30-minute periods spent hiking and there are 2 30-minute periods in every hour, it follows that the adult will need to hike for the fraction eight over 2, or 4 hours to use a total of 1,900 calories from bicycling and hiking.

Choice A is incorrect and may result from solving the equation 200 h equals 1,900. This represents 0 30-minute periods bicycling instead of 2. Choice B is incorrect and may result from solving the equation 200 h plus 150, equals 1,900. This represents 1 30-minute period of bicycling instead of 2. Choice C is incorrect. This may result from determining that the number of 30-minute periods the adult should hike is 8, but then subtracting 2 from 8, rather than dividing 8 by 2, to find the number of hours the adult should hike. 

 

Question 334 334 of 569 selected Systems Of 2 Linear Equations In 2 Variables M

y=3x+9

3y=8x-6

The solution to the given system of equations is (x,y). What is the value of x - y ?

  1. -123

  2. -33

  3. 3

  4. 57

Show Answer Correct Answer: D

Choice D is correct. The first equation in the given system of equations defines y as 3x+9. Substituting 3x+9 for y in the second equation in the given system of equations yields 3(3x+9)=8x-6. Applying the distributive property on the left-hand side of this equation yields 9x+27=8x-6. Subtracting 8x from both sides of this equation yields x+27=-6. Subtracting 27 from both sides of this equation yields x=-33. Substituting -33 for x in the first equation of the given system of equations yields y=3(-33)+9, or y=-90. Substituting -33 for x and -90 for y into the expression x-y yields -33-(-90), or 57.

Choice A is incorrect. This is the value of x+y, not x-y.

Choice B is incorrect. This is the value of x, not x-y.

Choice C is incorrect and may result from conceptual or calculation errors.

Question 335 335 of 569 selected Linear Functions M

In the linear function h h(28)=15 and h(26)=22. Which equation defines h ?

  1. h(x)=-27x+23

  2. h(x)=-27x+113

  3. h(x)=-72x+23

  4. h(x)=-72x+113

Show Answer Correct Answer: D

Choice D is correct. An equation defining h can be written in the form y=mx+b, where y=h(x), m represents the slope of the graph of y=h(x) in the xy-plane, and b represents the y-coordinate of the y-intercept of the graph. It’s given that h(28)=15 and h(26)=22. It follows that the points (28,15) and (26,22) are on the graph of y=h(x) in the xy-plane. The slope can be found by using any two points, (x1,y1) and (x2,y2), and the formula m=y2-y1x2-x1. Substituting (28,15) and (26,22) for (x1,y1) and (x2,y2), respectively, in the slope formula yields m=22-1526-28, or m = - 7 2 . Substituting - 7 2 for m and (28,15) for (x,y) in the equation y=mx+b yields 15=(-72)(28)+b, or 15=-98+b. Adding 98 to both sides of this equation yields 113=b. Substituting - 7 2 for m and 113 for b in the equation y=mx+b yields y=-72x+113. Since y=h(x), it follows that the equation that defines h is h(x)=-72x+113.

Choice A is incorrect. For this function, h(26)=1097, not h(26)=22.

Choice B is incorrect. For this function, h(28)=105, not h(28)=15, and h(26)=7397, not h(26)=22.

Choice C is incorrect. For this function, h(28)=-75, not h(28)=15, and h(26)=-68, not h(26)=22.

Question 336 336 of 569 selected Linear Functions M

The figure presents a 3-column table, with 3 rows of data, titled “Energy per Gram of Typical Macronutrients.” The heading for column 1 is “Macronutrient,” the heading for column 2 is “Food calories,” and the heading for column 3 is “Kilojoules.” The 3 rows of data are as follows. Row 1.Macronutrient, Protein; Food calories, 4 point zero; Kilojoules, 16 point 7. Row 2. Macronutrient, Fat; Food calories, 9 point zero; Kilojoules, 37 point 7. Row 3. Macronutrient, Carbohydrate; Food calories, 4 point zero; Kilojoules, 16 point 7.

The table above gives the typical amounts of energy per gram, expressed in both food calories and kilojoules, of the three macronutrients in food. If x food calories is equivalent to k kilojoules, of the following, which best represents the relationship between x and k ?

 

  1. k equals zero point 2 4 x

  2. k equals 4 point 2 x

  3. x equals 4 point 2 k

  4. x k equals 4 point 2

Show Answer Correct Answer: B

Choice B is correct. The relationship between x food calories and k kilojoules can be modeled as a proportional relationship. Let the ordered pair x sub 1 comma k sub 1 and the ordered pair x sub 2 comma k sub 2 represent the values in the first two rows in the table: the ordered pair 4 point 0 comma 16 point 7 and the ordered pair 9 point 0 comma 37 point 7. The rate of change, or the fraction with numerator, k sub 2 minus k sub 1, and denominator x sub 2 minus x sub 1, end fraction, is 21 over 5 equals 4 point 2; therefore, the equation that best represents the relationship between x and k is k equals 4 point 2 x.

Choice A is incorrect and may be the result of calculating the rate of change using the fraction with numerator, x sub 2 minus x sub 1, and denominator k sub 2 minus k sub 1, end fraction. Choice C is incorrect because the number of kilojoules is greater than the number of food calories. Choice D is incorrect and may be the result of an error when setting up the equation.

 

Question 337 337 of 569 selected Systems Of 2 Linear Equations In 2 Variables E

x = 4

y=5-x

The solution to the given system of equations is (x,y). What is the value of y ?

  1. 1

  2. 4

  3. 5

  4. 9

Show Answer Correct Answer: A

Choice A is correct. The first equation in the given system of equations is x=4. Substituting 4 for x in the second equation in the given system of equations yields y=5-4, or y=1.

Choice B is incorrect. This is the value of x in the solution to the given system of equations, not the value of y .

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 338 338 of 569 selected Systems Of 2 Linear Equations In 2 Variables M

y = 6 x + 3

One of the two equations in a system of linear equations is given. The system has infinitely many solutions. Which equation could be the second equation in this system?

  1. y=2(6x)+3

  2. y=2(6x+3)

  3. 2(y)=2(6x)+3

  4. 2(y)=2(6x+3)

Show Answer Correct Answer: D

Choice D is correct. It’s given that the system has infinitely many solutions. A system of two linear equations has infinitely many solutions when the two linear equations are equivalent. When one equation is a multiple of another equation, the two equations are equivalent. Multiplying each side of the given equation by 2 yields 2(y)=2(6x+3). Thus, 2(y)=2(6x+3) is equivalent to the given equation and could be the second equation in the system.

Choice A is incorrect. The system consisting of this equation and the given equation has one solution rather than infinitely many solutions.

Choice B is incorrect. The system consisting of this equation and the given equation has one solution rather than infinitely many solutions.

Choice C is incorrect. The system consisting of this equation and the given equation has no solutions rather than infinitely many solutions.

Question 339 339 of 569 selected Linear Inequalities In 1 Or 2 Variables H

I equals, V over R

The formula above is Ohm’s law for an electric circuit with current I, in amperes, potential difference V, in volts, and resistance R, in ohms. A circuit has a resistance of 500 ohms, and its potential difference will be generated by n six-volt batteries that produce a total potential difference of 6 n volts. If the circuit is to have a current of no more than 0.25 ampere, what is the greatest number, n, of six-volt batteries that can be used?

Show Answer

The correct answer is 20. For the given circuit, the resistance R is 500 ohms, and the total potential difference V generated by n batteries is 6 n volts. It’s also given that the circuit is to have a current of no more than 0.25 ampere, which can be expressed as I is less than 0 point 2 5. Since Ohm’s law says that I equals, V over R, the given values for V and R can be substituted for I in this inequality, which yields 6 n over 500 is less than 0 point 2 5. Multiplying both sides of this inequality by 500 yields 6 n is less than 125, and dividing both sides of this inequality by 6 yields n is less than 20 point 8 3 3. Since the number of batteries must be a whole number less than 20.833, the greatest number of batteries that can be used in this circuit is 20.

Question 340 340 of 569 selected Linear Equations In 1 Variable E

13x=112-x

What value of x is the solution to the given equation?

Show Answer Correct Answer: 8

The correct answer is 8 . Adding x to both sides of the given equation yields 14 x = 112 . Dividing both sides of this equation by 14 yields x = 8 .

Question 341 341 of 569 selected Linear Inequalities In 1 Or 2 Variables H

y<6x+2

For which of the following tables are all the values of x and their corresponding values of y solutions to the given inequality?

  1. x y
    3 20
    5 32
    7 44
  2. x y
    3 16
    5 36
    7 40
  3. x y
    3 16
    5 28
    7 40
  4. x y
    3 24
    5 36
    7 48
Show Answer Correct Answer: C

Choice C is correct. All the tables in the choices have the same three values of x , so each of the three values of x can be substituted in the given inequality to compare the corresponding values of y in each of the tables. Substituting 3 for x in the given inequality yields y<6(3)+2, or y<20. Therefore, when x = 3 , the corresponding value of y is less than 20 . Substituting 5 for x in the given inequality yields y<6(5)+2, or y<32. Therefore, when x = 5 , the corresponding value of y is less than 32 . Substituting 7 for x in the given inequality yields y<6(7)+2, or y<44. Therefore, when x = 7 , the corresponding value of y is less than 44 . For the table in choice C, when x = 3 , the corresponding value of y is 16 , which is less than 20 ; when x = 5 , the corresponding value of y is 28 , which is less than 32 ; when x = 7 , the corresponding value of y is 40 , which is less than 44 . Therefore, the table in choice C gives values of x and their corresponding values of y that are all solutions to the given inequality.

Choice A is incorrect. In the table for choice A, when x = 3 , the corresponding value of y is 20 , which is not less than 20 ; when x = 5 , the corresponding value of y is 32 , which is not less than 32 ; when x = 7 , the corresponding value of y is 44 , which is not less than 44 .

Choice B is incorrect. In the table for choice B, when x = 5 , the corresponding value of y is 36 , which is not less than 32 .

Choice D is incorrect. In the table for choice D, when x = 3 , the corresponding value of y is 24 , which is not less than 20 ; when x = 5 , the corresponding value of y is 36 , which is not less than 32 ; when x = 7 , the corresponding value of y is 48 , which is not less than 44 .

Question 342 342 of 569 selected Linear Equations In 1 Variable M

If 4 x minus one half, equals negative 5, what is the value of 8 x minus 1 ?

  1. 2

  2. negative nine eighths

  3. negative five halves

  4. negative 10

Show Answer Correct Answer: D

Choice D is correct. Multiplying the given equation by 2 on each side yields 2 times, open parenthesis, 4 x minus one half, close parenthesis, equals, 2 times negative 5. Applying the distributive property, this equation can be rewritten as 2 times 4 x, minus, 2 times one half, equals, 2 times negative 5, or 8 x minus 1, equals negative 10.

Choices A, B, and C are incorrect and may result from calculation errors in solving the given equation for x and then substituting that value of x in the expression 8 x minus 1.

 

Question 343 343 of 569 selected Linear Functions M

The table gives the number of hours, h , of labor and a plumber’s total charge f(h), in dollars, for two different jobs.

h f(h)
1 155
3 285

There is a linear relationship between h and f(h). Which equation represents this relationship?

  1. f(h)=25h+130

  2. f(h)=130h+25

  3. f(h)=65h+90

  4. f(h)=90h+65

Show Answer Correct Answer: C

Choice C is correct. It's given that there is a linear relationship between a plumber's hours of labor, h , and the plumber's total charge f(h), in dollars. It follows that the relationship can be represented by an equation of the form f(h)=mh+b, where m is the rate of change of the function f and b is a constant. The rate of change of f can be calculated by dividing the difference in two values of f(h) by the difference in the corresponding values of h . Based on the values given in the table, the rate of change of f is 285-1553-1, or 65 . Substituting 65 for m in the equation f(h)=mh+b yields f(h)=65h+b. The value of b can be found by substituting any value of h and its corresponding value of f(h) for h and f(h), respectively, in this equation. Substituting 1 for h and 155 for f(h) yields 155=65(1)+b, or 155=65+b. Subtracting 65 from both sides of this equation yields 90 = b . Substituting 90 for b in the equation f(h)=65h+b yields f(h)=65h+90.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 344 344 of 569 selected Linear Functions M

For the linear function g , the graph of y=g(x) in the xy-plane has a slope of 2 and passes through the point (1,14). Which equation defines g ?

  1. g(x)=2x

  2. g(x)=2x+2

  3. g(x)=2x+12

  4. g(x)=2x+14

Show Answer Correct Answer: C

Choice C is correct. An equation defining a linear function can be written in the form g(x)=mx+b, where m is the slope and (0,b) is the y-intercept of the graph of y=g(x) in the xy-plane. It’s given that the graph of y=g(x) has a slope of 2 . Therefore, m = 2 . It’s also given that the graph of y=g(x) passes through the point (1,14). It follows that when x = 1 , g(x)=14. Substituting 1 for x , 14 for g(x), and 2 for m in the equation g(x)=mx+b yields 14=2(1)+b, or 14=2+b. Subtracting 2 from each side of this equation yields 12 = b . Therefore, b = 12 . Substituting 2 for m and 12 for b in the equation g(x)=mx+b yields g(x)=2x+12. Therefore, the equation that defines g is g(x)=2x+12.

Choice A is incorrect. For this function, the graph of y=g(x) in the xy-plane passes through the point (1,2), not (1,14).

Choice B is incorrect. For this function, the graph of y=g(x) in the xy-plane passes through the point (1,4), not (1,14).

Choice D is incorrect. For this function, the graph of y=g(x) in the xy-plane passes through the point (1,16), not (1,14).

Question 345 345 of 569 selected Linear Functions H

The cost of renting a large canopy tent for up to 10 days is $430 for the first day and $215 for each additional day. Which of the following equations gives the cost y , in dollars, of renting the tent for x days, where x is a positive integer and x10?

  1. y = 215 x + 215

  2. y = 430 x - 215

  3. y = 430 x + 215

  4. y = 215 x + 430

Show Answer Correct Answer: A

Choice A is correct. It’s given that the cost of renting a large canopy tent is $430 for the first day and $215 for each additional day for up to 10 days. For x days of renting the tent, the cost includes $430 for the first day and $215 for each of the (x-1) additional days. It follows that the cost y, in dollars, of renting the tent can be expressed as y=430+215(x-1), which is equivalent to y=430+215x-215, or y=215x+215. Therefore, the equation y=215x+215 gives the cost of renting the tent for x days, where x is a positive integer and x10.

Choice B is incorrect. This equation represents a situation where the cost of renting the tent for the first day is $215, not $430, and the cost for each additional day is $430, not $215.

Choice C is incorrect. This equation represents a situation where the cost of renting the tent for the first day is $645, not $430, and the cost for each additional day is $430, not $215.

Choice D is incorrect. This equation represents a situation where the cost of renting the tent for the first day is $645, not $430.

Question 346 346 of 569 selected Linear Equations In 1 Variable M

A museum rents tablets to visitors. The museum earns revenue of $14 for each tablet rented for the day. On Wednesday, the museum earned $406 in profit from renting tablets after paying daily expenses of $112 . How many tablets did the museum rent on Wednesday? (profit=total revenuetotal expenses)

Show Answer Correct Answer: 37

The correct answer is 37. It's given that the museum earns revenue of $14 for each tablet rented for the day. It's also given that on Wednesday, the museum earned $406 in profit from renting tablets after paying daily expenses of $112. Let x represent the number of tablets the museum rented on Wednesday. It follows that the total revenue can be represented by the expression 14x. Because profit=total revenue-total expenses, the equation 406=14x-112 represents this situation. Adding 112 to both sides of this equation yields 14x=518. Dividing both sides of this equation by 14 yields x=37. Therefore, the museum rented 37 tablets on Wednesday.

Question 347 347 of 569 selected Systems Of 2 Linear Equations In 2 Variables M

A wire with a length of 106 inches is cut into two parts. One part has a length of x inches, and the other part has a length of y inches. The value of x is 6 more than 4 times the value of y . What is the value of x ?

  1. 25

  2. 28

  3. 56

  4. 86

Show Answer Correct Answer: D

Choice D is correct. It's given that a wire with a length of 106 inches is cut into two parts. It's also given that one part has a length of x inches and the other part has a length of y inches. This can be represented by the equation x+y=106. It's also given that the value of x is 6 more than 4 times the value of y . This can be represented by the equation x=4y+6. Substituting 4y+6 for x in the equation x+y=106 yields 4y+6+y=106, or 5y+6=106. Subtracting 6 from each side of this equation yields 5y=100. Dividing each side of this equation by 5 yields y=20. Substituting 20 for y in the equation x=4y+6 yields x=4(20)+6, or x=86

Choice A is incorrect. This value represents less than half of the total length of 106 inches; however, x represents the length of the longer part of the wire, since it's given that the value of x is 6 more than 4 times the value of y .

Choice B is incorrect. This value represents less than half of the total length of 106 inches; however, x represents the length of the longer part of the wire, since it's given that the value of x is 6 more than 4 times the value of y .

Choice C is incorrect. This represents a part that is 6 more than the length of the other part, rather than 6 more than 4 times the length of the other part.

Question 348 348 of 569 selected Linear Inequalities In 1 Or 2 Variables H

y>13x-18

For which of the following tables are all the values of x and their corresponding values of y solutions to the given inequality?

Show Answer Correct Answer: D

Choice D is correct. All the tables in the choices have the same three values of x , so each of the three values of x can be substituted in the given inequality to compare the corresponding values of y in each of the tables. Substituting 3 for x in the given inequality yields y>13(3)-18, or y>21. Therefore, when x = 3 , the corresponding value of y is greater than 21 . Substituting 5 for x in the given inequality yields y>13(5)-18, or y>47. Therefore, when x = 5 , the corresponding value of y is greater than 47 . Substituting 8 for x in the given inequality yields y>13(8)-18, or y>86. Therefore, when x = 8 , the corresponding value of y is greater than 86 . For the table in choice D, when x = 3 , the corresponding value of y is 26 , which is greater than 21 ; when x = 5 , the corresponding value of y is 52 , which is greater than 47 ; when x = 8 , the corresponding value of y is 91 , which is greater than 86 . Therefore, the table in choice D gives values of x and their corresponding values of y that are all solutions to the given inequality.

Choice A is incorrect. In the table for choice A, when x = 3 , the corresponding value of y is 21 , which is not greater than 21 ; when x = 5 , the corresponding value of y is 47 , which is not greater than 47 ; when x = 8 , the corresponding value of y is 86 , which is not greater than 86 .

Choice B is incorrect. In the table for choice B, when x = 5 , the corresponding value of y is 42 , which is not greater than 47 ; when x = 8 , the corresponding value of y is 86 , which is not greater than 86 .

Choice C is incorrect. In the table for choice C, when x = 3 , the corresponding value of y is 16 , which is not greater than 21 ; when x = 5 , the corresponding value of y is 42 , which is not greater than 47 ; when x = 8 , the corresponding value of y is 81 , which is not greater than 86 .

Question 349 349 of 569 selected Systems Of 2 Linear Equations In 2 Variables H

5 x + 14 y = 45

10 x + 7 y = 27

The solution to the given system of equations is (x,y). What is the value of x y ?

Show Answer Correct Answer: 1.8, 9/5

The correct answer is 9 5 . Multiplying the first equation in the given system by 2 yields 10 x + 28 y = 90 . Subtracting the second equation in the given system, 10 x + 7 y = 27 , from 10 x + 28 y = 90 yields (10x+28y)-(10x+7y)=90-27, which is equivalent to 10x+28y-10x-7y=63, or 21 y = 63 . Dividing both sides of this equation by 21 yields y = 3 . The value of x can be found by substituting 3 for y in either of the two given equations. Substituting 3 for y in the equation 10 x + 7 y = 27 yields 10x+7(3)=27, or 10 x + 21 = 27 . Subtracting 21 from both sides of this equation yields 10 x = 6 . Dividing both sides of this equation by 10 yields x=610, or x = 3 5 . Therefore, the value of x y is (35)(3), or 95. Note that 9/5 and 1.8 are examples of ways to enter a correct answer.

Question 350 350 of 569 selected Systems Of 2 Linear Equations In 2 Variables M

The sum of a number x and 7 is twice as large as a number y . The number y is 3 less than the number x . Which system of equations describes this situation?

  1. x+7=2y

    y=x-3

  2. x+7=2y

    y=3-x

  3. 2(x+7)=y

    y=x-3

  4. 2(x+7)=y

    y=3-x

Show Answer Correct Answer: A

Choice A is correct. It's given that the sum of a number x and 7 is twice as large as a number y. This can be described by the equation x+7=2y. It’s also given that the number y is 3 less than the number x. This can be described by the equation y=x-3. Therefore, the system consisting of the equations x+7=2y and y=x-3 describes this situation.

Choice B is incorrect. The equation y=3-x describes a situation where the number y is x less than 3.

Choice C is incorrect. The equation 2(x+7)=y describes a situation where the number y is twice the sum of a number x and 7.

Choice D is incorrect. The equation 2(x+7)=y describes a situation where the number y is twice the sum of a number x and 7, and the equation y=3-x describes a situation where a number y is x less than 3.

Question 351 351 of 569 selected Linear Inequalities In 1 Or 2 Variables E

Normal body temperature for an adult is between 97 point 8 degrees Fahrenheit and 99 degrees Fahrenheit, inclusive. If Kevin, an adult male, has a body temperature that is considered to be normal, which of the following could be his body temperature?

  1. 96 point 7 degrees Fahrenheit

  2. 97 point 6 degrees Fahrenheit

  3. 97 point 9 degrees Fahrenheit

  4. 99 point 7 degrees Fahrenheit

Show Answer Correct Answer: C

Choice C is correct. Normal body temperature must be greater than or equal to 97.8°F but less than or equal to 99°F. Of the given choices, 97.9°F is the only temperature that fits these restrictions.

Choices A and B are incorrect. These temperatures are less than 97.8°F, so they don’t fit the given restrictions. Choice D is incorrect. This temperature is greater than 99°F, so it doesn’t fit the given restrictions.

 

Question 352 352 of 569 selected Linear Equations In 2 Variables M

A total of 2 squares each have side length r . A total of 6 equilateral triangles each have side length t . None of these squares and triangles shares a side. The sum of the perimeters of all these squares and triangles is 210 . Which equation represents this situation?

  1. 6 r + 24 t = 210

  2. 2 r + 6 t = 210

  3. 8 r + 18 t = 210

  4. 6 r + 2 t = 210

Show Answer Correct Answer: C

Choice C is correct. It’s given that a total of 2 squares each have side length r . Therefore, each of the squares has perimeter 4 r . Since there are a total of 2 squares, the sum of the perimeters of these squares is 4r+4r, which is equivalent to 2(4r), or 8 r . It’s also given that a total of 6 equilateral triangles each have side length t . Therefore, each of the equilateral triangles has perimeter 3 t . Since there are a total of 6 equilateral triangles, the sum of the perimeters of these triangles is 3t+3t+3t+3t+3t+3t, which is equivalent to 6(3t), or 18 t . Since the sum of the perimeters of the squares is 8 r and the sum of the perimeters of the triangles is 18 t , the sum of the perimeters of all these squares and triangles is 8r+18t. It’s given that the sum of the perimeters of all these squares and triangles is 210 . Therefore, the equation 8 r + 18 t = 210 represents this situation.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 353 353 of 569 selected Linear Equations In 2 Variables H

The graph of 9x-10y=19 is translated down 4 units in the xy-plane. What is the x-coordinate of the x-intercept of the resulting graph?

Show Answer Correct Answer: 59/9, 6.555, 6.556

The correct answer is 59 9 . When the graph of an equation in the form A x + B y = C , where A , B , and C are constants, is translated down k units in the xy-plane, the resulting graph can be represented by the equation Ax+B(y+k)=C. It’s given that the graph of 9 x - 10 y = 19 is translated down 4 units in the xy-plane. Therefore, the resulting graph can be represented by the equation 9x-10(y+4)=19, or 9x-10y-40=19. Adding 40 to both sides of this equation yields 9 x - 10 y = 59 . The x-coordinate of the x-intercept of the graph of an equation in the xy-plane is the value of x in the equation when y = 0 . Substituting 0 for y in the equation 9 x - 10 y = 59 yields 9x-10(0)=59, or 9 x = 59 . Dividing both sides of this equation by 9 yields x = 59 9 . Therefore, the x-coordinate of the x-intercept of the resulting graph is 59 9 . Note that 59/9, 6.555, and 6.556 are examples of ways to enter a correct answer.

Question 354 354 of 569 selected Linear Equations In 2 Variables E
 
Characteristics for Rock Types
Rock typeWeight per volume (lb/ft3)Cost per pound
Basalt180$0.18
Granite165$0.09
Limestone120$0.03
Sandstone135$0.22

A city is planning to build a rock retaining wall, a monument, and a garden in a park. The table above shows four rock types that will be considered for use in the project. Also shown for each rock type is its weight per volume, in pounds per cubic foot (lb/ft3), and the cost per pound, in dollars. The equation 0 point 0 3 times, open parenthesis, 120 w, close parenthesis, plus, 0 point 1 8 times, open parenthesis, 180 z, close parenthesis, plus 3,385 point 8 0, equals 7,576 point 2 0. gives the total cost, in dollars, of the rocks used in the project in terms of the number of ft3 of limestone, w, and the number of ft3 of basalt, z. All four rock types are used in the project. Which of the following is the best interpretation of 3,385.80 in this context?

  1. The cost of the granite and sandstone needed for the project

  2. The cost of the basalt and limestone needed for the project

  3. The cost of the basalt needed for the project

  4. The cost of the sandstone needed for the project

Show Answer Correct Answer: A

Choice A is correct. The table shows the cost of limestone is $0.03/lb, and the weight per volume for limestone is 120 lb/ft3. Therefore, the term 0 point 0 3 times 120 w represents the cost, in dollars, of w ft3 of limestone. Similarly, the term 0 point 1 8 times 180 z represents the cost, in dollars, of z ft3 of basalt. The given equation shows that the total cost of all the rocks used in the project is $7,576.20. Since it’s given that all four rock types are used in the project, the remaining term, 3,385.80, represents the cost, in dollars, of the granite and sandstone needed for the project.

Choice B is incorrect. The cost of basalt and limestone needed for the project can be represented by 0 point 1 8 times 180 z, plus, 0 point 0 3 times 120 w. Choice C is incorrect. The cost of the basalt needed for the project can be represented by the expression 0 point 1 8 times 180 z. Choice D is incorrect and may result from neglecting to include granite in the rock types used for the project.

Question 355 355 of 569 selected Linear Equations In 1 Variable H

-3x+21px=84

In the given equation, p is a constant. The equation has no solution. What is the value of p ?

  1. 0

  2. 1 7

  3. 4 3

  4. 4

Show Answer Correct Answer: B

Choice B is correct. A linear equation in one variable has no solution if and only if the equation is false; that is, when there is no value of x that produces a true statement. It's given that in the equation -3x+21px=84, p is a constant and the equation has no solution for x . Therefore, the value of the constant p is one that results in a false equation. Factoring out the common factor of - 3 x on the left-hand side of the given equation yields -3x(1-7p)=84. Dividing both sides of this equation by -3 yields x(1-7p)=-28. Dividing both sides of this equation by (1-7p) yields x=-281-7p. This equation is false if and only if 1-7p=0. Adding 7 p to both sides of 1-7p=0 yields 1=7p. Dividing both sides of this equation by 7 yields 17=p. It follows that the equation x=-281-7p is false if and only if p=17. Therefore, the given equation has no solution if and only if the value of p is 17.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 356 356 of 569 selected Linear Inequalities In 1 Or 2 Variables E

Which of the following ordered pairs x comma y satisfies the inequality 5 x minus 3 y is less than 4 ?

  1. Statement 1, 1 comma 1
  2. Statement 2, 2 comma 5
  3. Statement 3, 3 comma 2​​​​​​​

  1. I only

  2. II only

  3. I and II only

  4. I and III only

Show Answer Correct Answer: C

Choice C is correct. Substituting the ordered pair 1 comma 1 into the inequality gives 5 times 1, minus, 3 times 1, is less than 4, or 2 is less than 4, which is a true statement. Substituting the ordered pair 2 comma 5 into the inequality gives 5 times 2, minus, 3 times 5, is less than 4, or negative 5 is less than 4, which is a true statement. Substituting the ordered pair 3 comma 2 into the inequality gives 5 times 3, minus, 3 times 2, is less than 4, or 9 is less than 4, which is not a true statement. Therefore, the ordered pair 1 comma 1 and the ordered pair 2 comma 5 are the only ordered pairs shown that satisfy the given inequality.

Choice A is incorrect because the ordered pair 2 comma 5 also satisfies the inequality. Choice B is incorrect because the ordered pair 1 comma 1 also satisfies the inequality. Choice D is incorrect because the ordered pair 3 comma 2 does not satisfy the inequality.

 

Question 357 357 of 569 selected Systems Of 2 Linear Equations In 2 Variables M

6 x + 7 y = 28
2 x + 2 y = 10

The solution to the given system of equations is (x,y). What is the value of y?

  1. -2

  2. 7

  3. 14

  4. 18

Show Answer Correct Answer: A

Choice A is correct. The given system of linear equations can be solved by the elimination method. Multiplying each side of the second equation in the given system by 3 yields (2x+2y)(3)=(10)(3), or 6x+6y=30. Subtracting this equation from the first equation in the given system yields (6x+7y)-(6x+6y)=(28)-(30), which is equivalent to (6x-6x)+(7y-6y)=28-30, or y=-2

Choice B is incorrect. This is the value of x , not the value of y .

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

 

Question 358 358 of 569 selected Linear Inequalities In 1 Or 2 Variables M

A particular botanist classifies a species of plant as tall if its typical height when fully grown is more than 100 centimeters. Each of the following inequalities represents the possible heights h , in centimeters, for a specific plant species when fully grown. Which inequality represents the possible heights h , in centimeters, for a tall plant species? 

  1. 106<h<158

  2. 80<h<100

  3. 42<h<87

  4. 17<h<85

Show Answer Correct Answer: A

Choice A is correct. It's given that a particular botanist classifies a species of plant as tall if its typical height when fully grown is more than 100 centimeters. The inequality 106<h<158 represents possible heights h, in centimeters, for a plant species when fully grown where h is between 106 and 158 centimeters. Since all values of h in this inequality are greater than 100 centimeters, this inequality represents the possible heights for a tall plant species.

Choice B is incorrect. This inequality represents possible heights h, in centimeters, for a plant species when fully grown where h is between 80 and 100 centimeters, not more than 100 centimeters.

Choice C is incorrect. This inequality represents possible heights h, in centimeters, for a plant species when fully grown where h is between 42 and 87 centimeters, not more than 100 centimeters.

Choice D is incorrect. This inequality represents possible heights h, in centimeters, for a plant species when fully grown where h is between 17 and 85 centimeters, not more than 100 centimeters.

Question 359 359 of 569 selected Linear Equations In 2 Variables E

The equation 46=2x+2y gives the perimeter of a rectangular rug that has length x, in feet, and width y, in feet. The width of the rug is 8 feet. What is the length, in feet, of the rug?

Show Answer Correct Answer: 15

The correct answer is 15 . It's given that the equation 46 = 2 x + 2 y gives the perimeter of a rectangular rug that has length x , in feet, and width y , in feet. It's also given that the width of the rug is 8 feet. Substituting 8 for y in the equation 46 = 2 x + 2 y yields 46=2x+2(8), or 46 = 2 x + 16 . Subtracting 16 from both sides of this equation yields 30 = 2 x . Dividing both sides of this equation by 2 yields 15 = x . Since x represents the length, in feet, of the rug, it follows that the length of the rug is 15 feet.

Question 360 360 of 569 selected Linear Functions E

  • The line slants gradually up from left to right.
  • The line passes through the following points:
    • (0 comma 20)
    • (5 comma 30)
    • (10 comma 40)

A bank account was opened with an initial deposit. Over the next several months, regular deposits were made into this account, and there were no withdrawals made during this time. The graph of the function f shown, where y=f(x), estimates the account balance, in dollars, in this bank account x months since the initial deposit. To the nearest whole dollar, what is the amount of the initial deposit estimated by the graph?

Show Answer Correct Answer: 20

The correct answer is 20 . For the graph shown, the x-axis represents the time since the initial deposit, in months, and the y-axis represents the bank account balance, in dollars. The amount of the initial deposit is estimated by the y-coordinate of the point on the graph that represents 0 months since the initial deposit. Therefore, the amount of the initial deposit is estimated by the corresponding y-value for the point when x = 0 . When x = 0 , it is estimated that y = 20 . Thus, the amount of the initial deposit estimated by the graph, to the nearest whole dollar, is 20 .

Question 361 361 of 569 selected Linear Equations In 2 Variables M

When line n is graphed in the xy-plane, it has an x-intercept of (-4,0) and a y-intercept of (0,863). What is the slope of line n ?

  1. 3 344

  2. 6 43

  3. 43 6

  4. 344 3

Show Answer Correct Answer: C

Choice C is correct. It's given that when line n is graphed in the xy-plane, it has an x-intercept of (-4,0) and a y-intercept of (0,863). The slope, m , of a line can be found using any two points on the line, (x1,y1) and (x2,y2), and the slope formula m=y2-y1x2-x1. Substituting the points (-4,0) and (0,863) for (x1,y1) and (x2,y2), respectively, in the slope formula yields m=863-00-(-4), or m=436. Therefore, the slope of line n is 436.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect. This is the slope of a line that has an x-intercept of (863,0) and a y-intercept of (0,-4).

Choice D is incorrect and may result from conceptual or calculation errors.

Question 362 362 of 569 selected Systems Of 2 Linear Equations In 2 Variables E

A dance teacher ordered outfits for students for a dance recital. Outfits for boys cost $26, and outfits for girls cost $35. The dance teacher ordered a total of 28 outfits and spent $881. If b represents the number of outfits the dance teacher ordered for boys and g represents the number of outfits the dance teacher ordered for girls, which of the following systems of equations can be solved to find b and g ?

  1. Each option consists of two equations. 26 b plus 35 g, equals 28, 
and, b plus g, equals 881

  2. 26 b plus 35 g, equals 881, 
and,
b plus g, equals 28

  3. 26 g plus 35 b, equals 28, and, b plus g, equals 881

  4. 26 g plus 35 b, equals 881, and, b plus g, equals 28

Show Answer Correct Answer: B

Choice B is correct. Outfits for boys cost $26 each and the teacher ordered b outfits for boys, so the teacher spent 26b dollars on outfits for boys. Similarly, outfits for girls cost $35 each and the teacher ordered g outfits for girls, so the teacher spent 35g dollars on outfits for girls. Since the teacher spent a total of $881 on outfits for boys and girls, the equation 26b + 35g = 881 must be true. And since the teacher ordered a total of 28 outfits, the equation b + g = 28 must also be true.


Choice A is incorrect and may result from switching the constraint on the total number of outfits with the constraint on the cost of the outfits. Choice C is incorrect and may result from switching the constraint on the total number of outfits with the constraint on the cost of the outfits, as well as switching the cost of the outfits for boys with the cost of the outfits for girls. Choice D is incorrect and may result from switching the cost of the outfits for boys with the cost of the outfits for girls.

 

Question 363 363 of 569 selected Systems Of 2 Linear Equations In 2 Variables M

3 x + 6 = 4 y

3 x + 4 = 2 y

The solution to the given system of equations is (x,y). What is the value of y ?

Show Answer Correct Answer: 1

The correct answer is 1 . Subtracting the second equation from the first equation in the given system of equations yields (3x-3x)+(6-4)=4y-2y, which is equivalent to 0+2=2y, or 2=2y. Dividing each side of this equation by 2 yields 1 = y .

Question 364 364 of 569 selected Systems Of 2 Linear Equations In 2 Variables M

y = 2 7 x + 3

One of the two equations in a system of linear equations is given. The system has infinitely many solutions. If the second equation in the system is y = m x + b , where m and b are constants, what is the value of b ?

  1. -3

  2. - 1 3

  3. 1 3

  4. 3

Show Answer Correct Answer: D

Choice D is correct. It’s given that the system has infinitely many solutions. The graphs of two lines in the xy-plane represented by equations in slope-intercept form, y=mx+b, where m and b are constants, have infinitely many solutions if their slopes, m, are the same and if their y-coordinates of the y-intercepts, b, are also the same. The first equation in the given system is y=27x+3. For this equation, the slope is 27 and the y-coordinate of the y-intercept is 3. If the second equation is in the form y=mx+b, then for the two equations to be equivalent, the values of m and b in the second equation must equal the corresponding values in the first equation. Therefore, the second equation must have a slope, m, of 27, and a y-coordinate of the y-intercept, b, of 3. Thus, the value of b is 3.

Choice A is incorrect and may result from conceptual errors.

Choice B is incorrect and may result from conceptual errors.

Choice C is incorrect and may result from conceptual errors.

Question 365 365 of 569 selected Linear Functions M

The function f is defined by f(x)=x+155, and f(a)=10, where a is a constant. What is the value of a ?

  1. 5

  2. 10

  3. 35

  4. 65

Show Answer Correct Answer: C

Choice C is correct. It's given that f(x)=x+155 and f(a)=10, where a is a constant. Therefore, for the given function f , when x = a f(x)=10. Substituting a for x and 10 for f(x) in the given function f yields 10=a+155. Multiplying both sides of this equation by 5 yields 50=a+15. Subtracting 15 from both sides of this equation yields 35=a. Therefore, the value of a is 35 .

Choice A is incorrect. This is the value of a if f(a)=4.

Choice B is incorrect. This is the value of a if f(a)=5.

Choice D is incorrect. This is the value of a if f(a)=16.

Question 366 366 of 569 selected Systems Of 2 Linear Equations In 2 Variables H

Equation 1: y equals, one half x, plus 8. Equation 2: y equals, c x plus 10.

In the system of equations above, c is a constant. If the system has no solution, what is the value of c ?

Show Answer

The correct answer is one half. A system of two linear equations has no solution when the graphs of the equations have the same slope and different y-intercepts. Each of the given linear equations is written in the slope-intercept form, y equals, m x plus b, where m is the slope and b is the y-coordinate of the y-intercept of the graph of the equation. For these two linear equations, the y-intercepts are the points with coordinates 0 comma 8  and 0 comma 10 . Thus, if the system of equations has no solution, the slopes of the graphs of the two linear equations must be the same. The slope of the graph of the first linear equation is one half. Therefore, for the system of equations to have no solution, the value of c must be one half. Note that 1/2 and .5 are examples of ways to enter a correct answer.

Question 367 367 of 569 selected Linear Equations In 1 Variable M

If 2(3t-10)+t=40+4t, what is the value of 3t?

Show Answer Correct Answer: 60

The correct answer is 60 . Subtracting t from both sides of the given equation yields 2(3t-10)=40+3t. Applying the distributive property to the left-hand side of this equation yields 6t-20=40+3t. Adding 20 to both sides of this equation yields 6t=60+3t. Subtracting 3 t from both sides of this equation yields 3t=60. Therefore, the value of 3 t is 60 .

Question 368 368 of 569 selected Linear Equations In 1 Variable E

If 6 n = 12 , what is the value of n + 4 ?

Show Answer Correct Answer: 6

The correct answer is 6 . Dividing both sides of the equation 6 n = 12 by 6 yields n = 2 . Substituting 2 for n in the expression n + 4 yields 2+4, or 6 .

Question 369 369 of 569 selected Linear Equations In 1 Variable H

If x+63=x+613, the value of x+6 is between which of the following pairs of values?

  1. -7 and -3

  2. -2 and 2

  3. 2 and 7

  4. 8 and 13

Show Answer Correct Answer: B

Choice B is correct. Multiplying both sides of the given equation by (3)(13), or 39 , yields (39)(x+63)=(39)(x+613), or 13(x+6)=3(x+6). Subtracting 3(x+6) from both sides of this equation yields 10(x+6)=0. Dividing both sides of this equation by 10 yields x+6=0. Therefore, if x+63=x+613, then the value of x+6 is 0 . It follows that of the given choices, the value of x+6 is between -2 and 2 .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 370 370 of 569 selected Linear Inequalities In 1 Or 2 Variables M

A city employee will plant two types of bushes, azaleas and boxwoods, in a park. There will be no more than 164 total bushes planted, and the number of azaleas planted will be at most three times the number of boxwoods planted. Which of the following systems of inequalities best represents this situation, where a is the number of azaleas that will be planted, and b is the number of boxwoods that will be planted?

  1. a+b164

    3ab

  2. a+b164

    a3b

  3. a+b164

    3ab

  4. a+b164

    a3b

Show Answer Correct Answer: D

Choice D is correct. It’s given that a city employee will plant azaleas and boxwoods in a park, where a is the number of azaleas that will be planted and b is the number of boxwoods that will be planted. It’s also given that there will be no more than 164 total bushes planted, which can be represented by the inequality a+b164. It’s also given that the number of azaleas planted will be at most three times the number of boxwoods planted, which can be represented by the inequality a3b. Therefore, the system of inequalities containing a+b164 and a3b best represents this situation.

Choice A is incorrect. The inequality a+b164 represents a situation where at least 164 total bushes will be planted, not that there will be no more than 164 total bushes planted. Also, the inequality 3ab represents a situation where the number of boxwoods that will be planted is at most three times the number of azaleas that will be planted, not that the number of azaleas planted will be at most three times the number of boxwoods planted.

Choice B is incorrect. The inequality a+b164 represents a situation where at least 164 total bushes will be planted, not that there will be no more than 164 total bushes planted.

Choice C is incorrect. The inequality 3ab represents a situation where the number of boxwoods that will be planted is at most three times the number of azaleas that will be planted, not that the number of azaleas planted will be at most three times the number of boxwoods planted.

Question 371 371 of 569 selected Linear Inequalities In 1 Or 2 Variables E

A clothing store is having a sale on shirts and pants. During the sale, the cost of each shirt is $15 and the cost of each pair of pants is $25. Geoff can spend at most $120 at the store. If Geoff buys s shirts and p pairs of pants, which of the following must be true?

  1. 15 s plus 25 p, is less than or equal to 120

  2. 15 s plus 25 p, is greater than or equal to 120

  3. 25 s plus 15 p, is less than or equal to 120

  4. 25 s plus 15 p, is greater than or equal to 120

Show Answer Correct Answer: A

Choice A is correct. Since the cost of each shirt is $15 and Geoff buys s shirts, the expression 15 s represents the amount Geoff spends on shirts. Since the cost of each pair of pants is $25 and Geoff buys p pairs of pants, the expression 25 p represents the amount Geoff spends on pants. Therefore, the sum 15 s plus 25 p represents the total amount Geoff spends at the store. Since Geoff can spend at most $120 at the store, the total amount he spends must be less than or equal to 120. Thus, 15 s plus 25 p, is less than or equal to 120.

Choice B is incorrect. This represents the situation in which Geoff spends at least, rather than at most, $120 at the store. Choice C is incorrect and may result from reversing the cost of a shirt and that of a pair of paints. Choice D is incorrect and may result from both reversing the cost of a shirt and that of a pair of pants and from representing a situation in which Geoff spends at least, rather than at most, $120 at the store.

 

Question 372 372 of 569 selected Linear Equations In 1 Variable E

If 2 n over 5, equals 10, what is the value of 2 n, minus 1 ?

  1. 24

  2. 49

  3. 50

  4. 99

Show Answer Correct Answer: B

Choice B is correct. Multiplying both sides of the given equation by 5 yields 2 n equals 50. Substituting 50 for 2 n in the expression 2 n minus 1 yields 50 minus 1, equals 49.

Alternate approach: Dividing both sides of 2 n equals 50 by 2 yields n equals 25. Evaluating the expression 2 n minus 1 for n equals 25 yields 2 times 25, minus 1, equals 49.

Choice A is incorrect and may result from finding the value of n minus 1 instead of 2 n minus 1. Choice C is incorrect and may result from finding the value of 2 n instead of 2 n minus 1. Choice D is incorrect and may result from finding the value of 4 n minus 1 instead of 2 n minus 1.

Question 373 373 of 569 selected Linear Equations In 2 Variables M

In the xy-plane, line k passes through the points (0,-5) and (1,-1). Which equation defines line k ?

  1. y = - x + 1 4

  2. y=14x-5

  3. y = - x + 4

  4. y = 4 x - 5

Show Answer Correct Answer: D

Choice D is correct. An equation defining a line in the xy-plane can be written in the form y = m x + b , where m represents the slope and (0,b) represents the y-intercept of the line. It’s given that line k passes through the point (0,-5); therefore, b = -5 . The slope, m , of a line can be found using any two points on the line, (x1,y1) and (x2,y2), and the slope formula m=y2-y1x2-x1. Substituting the points (0,-5) and (1,-1) for (x1,y1) and (x2,y2), respectively, in the slope formula yields m=(-1-(-5))(1-0), or m = 4 . Substituting 4 for m and -5 for b in the equation y = m x + b yields y = 4 x - 5

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Question 374 374 of 569 selected Linear Equations In 2 Variables H
x y
3 7
k 11
12 n

The table above shows the coordinates of three points on a line in the xy-plane, where k and n are constants. If the slope of the line is 2, what is the value of k plus n ?

Show Answer

The correct answer is 30. The slope of a line can be found by using the slope formula, the fraction with numerator y sub 2 minus y sub 1, and denominator x sub 2 minus x sub 1, end fraction. It’s given that the slope of the line is 2; therefore, the fraction with numerator y sub 2 minus y sub 1, and denominator x sub 2 minus x sub 1, end fraction, equals 2. According to the table, the points with coordinates 3 comma 7 and with coordinates k comma 11 lie on the line. Substituting the coordinates of these points into the equation gives the fraction with numerator 11 minus 7, and denominator k minus 3, end fraction, equals 2. Multiplying both sides of this equation by k minus 3 gives 11 minus 7, equals, 2 times, open parenthesis, k minus 3, close parenthesis, or 11 minus 7, equals, 2 k minus 6. Solving for k gives k equals 5. According to the table, the points with coordinates 3 comma 7 and with coordinates 12 comma n also lie on the line. Substituting the coordinates of these points into the fraction with numerator y sub 2 minus y sub 1, and denominator x sub 2 minus x sub 1, end fraction, equals 2 gives the fraction with numerator n minus 7, and denominator 12 minus 3, end fraction, equals 2. Solving for n gives n equals 25. Therefore, k plus n, equals, 5 plus 25, or 30.

Question 375 375 of 569 selected Systems Of 2 Linear Equations In 2 Variables E

  • For the first line in the system:
    • The line slants gradually down from left to right.
    • The line passes through the following points:
      • (0 comma 5)
      • (2 comma 4)
      • (10 comma 0)
  • For the second line in the system:
    • The line slants sharply up from left to right.
    • The line passes through the following points:
      • (0 comma 0)
      • (2 comma 4)
      • (5 comma 10)

The graph of a system of linear equations is shown. What is the solution (x,y) to the system?

  1. (0,5)

  2. (2,4)

  3. (5,10)

  4. (10,0)

Show Answer Correct Answer: B

Choice B is correct. A solution to a system of equations must be the solution to each equation in the system. It follows that if (x,y) is a solution to the system, the point (x,y) lies on the graph in the xy-plane of each equation in the system. The point that lies on each graph of the system of linear equations shown is their intersection point (2,4). Therefore, the solution to the system is (2,4).

Choice A is incorrect. The point (0,5) lies on one, but not both, of the graphs of the linear equations shown.

Choice C is incorrect. The point (5,10) lies on one, but not both, of the graphs of the linear equations shown.

Choice D is incorrect. The point (10,0) lies on one, but not both, of the graphs of the linear equations shown.

Question 376 376 of 569 selected Systems Of 2 Linear Equations In 2 Variables H

7 2 x + 6 y = 25

5 2 x + 6 y = 23

The solution to the given system of equations is (x,y). What is the value of 17 2 x + 18 y ?

  1. 2

  2. 3

  3. 48

  4. 71

Show Answer Correct Answer: D

Choice D is correct. Multiplying the second equation in the given system by 2 yields 102x+12y=46. Adding this equation to the first equation in the system yields (72x+6y)+(102x+12y)=25+46, which is equivalent to (72x+102x)+(6y+12y)=25+46, or 172x+18y=71. Therefore, the value of 172x+18y is 71 .

Choice A is incorrect. This is the value of x , not the value of 172x+18y.

Choice B is incorrect. This is the value of y , not the value of 172x+18y.

Choice C is incorrect. This the value of (72x+6y)+(52x+6y), or 6 x + 12 y , not the value of 172x+18y.

Question 377 377 of 569 selected Linear Equations In 1 Variable E

2.6+x=2.8

What value of x is the solution to the given equation?

Show Answer Correct Answer: 0.2, 1/5

The correct answer is .2. Subtracting 2.6 from each side of the given equation yields x=0.2. Therefore, the value of x that's the solution to the given equation is 0.2. Note that .2 and 1/5 are examples of ways to enter a correct answer.

Question 378 378 of 569 selected Linear Functions E

The function f is defined by the equation f(x)=100x+2. What is the value of f(x) when x = 9 ?

  1. 111

  2. 118

  3. 900

  4. 902

Show Answer Correct Answer: D

Choice D is correct. Substituting 9 for x in the given equation yields f(9)=100(9)+2, or f(9)=902. Therefore, the value of f(x) when x=9 is 902 .

Choice A is incorrect. This is the value of f(x) when x=1.09.

Choice B is incorrect. This is the value of f(x) when x=1.16.

Choice C is incorrect. This is the value of f(x) when x=8.98.

Question 379 379 of 569 selected Linear Equations In 2 Variables E

A line in the xy-plane has a slope of 19 and passes through the point (0,14). Which equation represents this line?

  1. y=-19x-14

  2. y=-19x+14

  3. y=19x-14

  4. y=19x+14

Show Answer Correct Answer: D

Choice D is correct. The equation of a line in the xy-plane can be written as y=mx+b, where m represents the slope of the line and (0,b) represents the y-intercept of the line. It's given that the slope of the line is 19. It follows that m=19. It's also given that the line passes through the point (0,14). It follows that b=14. Substituting 19 for m and 14 for b in y=mx+b yields y=19x+14. Thus, the equation y=19x+14 represents this line.

Choice A is incorrect. This equation represents a line with a slope of -19 and a y-intercept of (0,-14).

Choice B is incorrect. This equation represents a line with a slope of -19 and a y-intercept of (0,14).

Choice C is incorrect. This equation represents a line with a slope of 19 and a y-intercept of (0,-14).

Question 380 380 of 569 selected Systems Of 2 Linear Equations In 2 Variables E

s+7r=27

r=3

What is the solution (r,s) to the given system of equations? 

  1. (6,3)

  2. (3,6)

  3. (3,27)

  4. (27,3)

Show Answer Correct Answer: B

Choice B is correct. The second equation in the given system is r = 3 . Substituting 3 for r in the first equation in the given system yields s+7(3)=27, or s + 21 = 27 . Subtracting 21 from both sides of this equation yields s = 6 . Therefore, the solution (r,s) to the given system of equations is (3,6).

Choice A is incorrect. This is the solution (s,r), not (r,s), to the given system of equations.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 381 381 of 569 selected Linear Equations In 2 Variables E

Vivian bought party hats and cupcakes for $71 . Each package of party hats cost $3 , and each cupcake cost $1 . If Vivian bought 10 packages of party hats, how many cupcakes did she buy?

Show Answer Correct Answer: 41

The correct answer is 41 . The number of cupcakes Vivian bought can be found by first finding the amount Vivian spent on cupcakes. The amount Vivian spent on cupcakes can be found by subtracting the amount Vivian spent on party hats from the total amount Vivian spent. The amount Vivian spent on party hats can be found by multiplying the cost per package of party hats by the number of packages of party hats, which yields $3·10, or $30. Subtracting the amount Vivian spent on party hats, $30, from the total amount Vivian spent, $71, yields $71-$30, or $41. Since the amount Vivian spent on cupcakes was $41 and each cupcake cost $1, it follows that Vivian bought 41 cupcakes.

Question 382 382 of 569 selected Linear Equations In 2 Variables E

  • The line slants sharply up from left to right.
  • The line passes through the following points:
    • (0 comma negative 5)
    • (1 comma negative 3)
    • (2 comma negative 1)

The graph shows the linear relationship between x and y . Which table gives three values of x and their corresponding values of y for this relationship?

Show Answer Correct Answer: D

Choice D is correct. It’s given that the graph shows the linear relationship between x and y . The given graph passes through the points (0,-5), (1,-3), and (2,-1). It follows that when x = 0 , the corresponding value of y is -5 , when x = 1 , the corresponding value of y is -3 , and when x = 2 , the corresponding value of y is -1 . Of the given choices, only the table in choice D gives these three values of x and their corresponding values of y for the relationship shown in the graph.

Choice A is incorrect. This table represents a relationship between x and y such that the graph passes through the points (0,0), (1,-7), and (2,-9).

Choice B is incorrect. This table represents a relationship between x and y such that the graph passes through the points (0,0), (1,-3), and (2,-1).

Choice C is incorrect. This table represents a linear relationship between x and y such that the graph passes through the points (0,-5), (1,-7), and (2,-9).

Question 383 383 of 569 selected Linear Equations In 2 Variables H

  • The line slants gradually down from left to right.
  • The line passes through the following points:
    • (negative 6 comma 0)
    • (0 comma negative 5)

Line k is shown in the xy-plane. Line j (not shown) is perpendicular to line k . What is the slope of line j ?

Show Answer Correct Answer: 1.2, 6/5

The correct answer is 65. It’s given that line j is perpendicular to line k in the xy-plane. This means that the slope of line j is the opposite reciprocal of the slope of line k. For a line that passes through the points (x1,y1) and (x2,y2) in the xy-plane, the slope of the line can be calculated as y2-y1x2-x1. It's shown that line k passes through the points (-6,0) and (0,-5) in the xy-plane. Substituting -6 for x1, 0 for y1, 0 for x2, and -5 for y2 in y2-y1x2-x1 yields -5-00-(-6), or -56. The opposite reciprocal of -56 is 65. Therefore, the slope of line j is 65. Note that 6/5 and 1.2 are examples of ways to enter a correct answer.

Question 384 384 of 569 selected Linear Inequalities In 1 Or 2 Variables M

The length of a rectangle is 50 inches and the width is x inches. The perimeter is at most 210 inches. Which inequality represents this situation?

  1. 2x+100210

  2. 2x+100210

  3. 2x+50210

  4. 2x+50210

Show Answer Correct Answer: A

Choice A is correct. The perimeter of a rectangle is equal to the sum of 2 times its length and 2 times its width. It's given that the rectangle's length is 50 inches and the width is x inches. Therefore, the perimeter, in inches, is 2(50)+2x, or 100+2x, which is equivalent to 2 x + 100 . It's given that the perimeter is at most 210 inches; therefore, 2x+100210 represents this situation.

Choice B is incorrect. This inequality represents a situation where the perimeter is at least, rather than at most, 210 inches.

Choice C is incorrect. This inequality represents a situation where 2 times the length, rather than the length, is 50 inches.

Choice D is incorrect. This inequality represents a situation where 2 times the length, rather than the length, is 50 inches, and the perimeter is at least, rather than at most, 210 inches.

Question 385 385 of 569 selected Linear Inequalities In 1 Or 2 Variables H

A shipping service restricts the dimensions of the boxes it will ship for a certain type of service. The restriction states that for boxes shaped like rectangular prisms, the sum of the perimeter of the base of the box and the height of the box cannot exceed 130 inches. The perimeter of the base is determined using the width and length of the box. If a box has a height of 60 inches and its length is 2.5 times the width, which inequality shows the allowable width x, in inches, of the box?

  1. zero is less than x, which is less than or equal to 10

  2. zero is less than x, which is less than or equal to 11 and two-thirds

  3. zero is less than x, which is less than or equal to 17 and one-half

  4. zero is less than x, which is less than or equal to 20

Show Answer Correct Answer: A

Choice A is correct. If x is the width, in inches, of the box, then the length of the box is 2.5x inches. It follows that the perimeter of the base is 2 times, open parenthesis, 2 point 5 x plus x, close parenthesis, or 7x inches. The height of the box is given to be 60 inches. According to the restriction, the sum of the perimeter of the base and the height of the box should not exceed 130 inches. Algebraically, this can be represented by 7 x plus 60, is less than or equal to 130, or 7 x is less than or equal to 70. Dividing both sides of the inequality by 7 gives x is less than or equal to 10. Since x represents the width of the box, x must also be a positive number. Therefore, the inequality 0 is less than x, which is less than or equal to 10 represents all the allowable values of x that satisfy the given conditions.

Choices B, C, and D are incorrect and may result from calculation errors or misreading the given information.

 

Question 386 386 of 569 selected Linear Equations In 2 Variables H

Line h is defined by 15x+17y-70=0. Line j is perpendicular to line h in the xy-plane. What is the slope of line j ?

  1. -75

  2. -57

  3. 75

  4. 57

Show Answer Correct Answer: D

Choice D is correct. It’s given that line h is defined by 15x+17y-70=0. This equation can be written in slope-intercept form y=mx+b, where m is the slope of line h and b is the y-coordinate of the y-intercept of line h . Adding 70 to both sides of 15x+17y-70=0 yields 15x+17y=70. Subtracting 15x from both sides of this equation yields 17y=-15x+70. Multiplying both sides of this equation by 7 yields y=-75x+490. Therefore, the slope of line h is -75. It’s given that line j is perpendicular to line h in the xy-plane. Two lines are perpendicular if their slopes are negative reciprocals, meaning that the slope of the first line is equal to - 1 divided by the slope of the second line. Therefore, the slope of line j is the negative reciprocal of the slope of line h . The negative reciprocal of -75 is -1(-75), or 57. Therefore, the slope of line j is 57.

Choice A is incorrect. This is the slope of a line in the xy-plane that is parallel, not perpendicular, to line h .

Choice B is incorrect. This is the reciprocal, not the negative reciprocal, of -75.

Choice C is incorrect. This is the negative, not the negative reciprocal, of -75.

Question 387 387 of 569 selected Linear Equations In 1 Variable M

The cost to rent a commercial fishing boat from a certain company is $950 for the first 2 hours and an additional $50 per hour for each hour after the first 2 hours. If the total cost to rent the commercial fishing boat from the company for t hours, where t>2, is $1,100, which equation represents this situation?

  1. 950(t-2)+50t=1,100

  2. 950(2t)+50t=1,100

  3. 950+50(t-2)=1,100

  4. 950+50(2t)=1,100

Show Answer Correct Answer: C

Choice C is correct. It’s given that the cost to rent a commercial fishing boat is $950 for the first 2 hours and an additional $50 per hour for each hour after the first 2 hours. It’s also given that t represents the total number of hours and t>2. Therefore, the number of additional hours after the first 2 hours can be represented with the expression t-2. The cost for these additional hours is $50 per hour, so the cost for the additional hours can be represented by the expression 50(t-2). The total cost can be calculated by adding the cost for the first 2 hours to the cost for the additional hours and can be represented by the expression 950+50(t-2). It’s also given that the total cost to rent the commercial fishing boat from the company for t hours is $1,100. Thus, the equation that represents this situation is 950+50(t-2)=1,100.

Choice A is incorrect and may result from conceptual errors.

Choice B is incorrect and may result from conceptual errors.

Choice D is incorrect and may result from conceptual errors.

Question 388 388 of 569 selected Linear Equations In 2 Variables E
x y
1 5
2 7
3 9
4 11

The table above shows some pairs of x values and y values. Which of the following equations could represent the relationship between x and y ?

  1. y equals 2 x plus 3

  2. y equals 3 x minus 2

  3. y equals 4 x minus one

  4. y equals 5 x

Show Answer Correct Answer: A

Choice A is correct. Each of the choices is a linear equation in the form y = mx + b, where m and b are constants. In this equation, m represents the change in y for each increase in x by 1. From the table, it can be determined that the value of y increases by 2 for each increase in x by 1. In other words, for the  pairs of x and y in the given table, m = 2. The value of b can be found by substituting the values of x and y from any row of the table and substituting the value of m into the equation y = mx + b and then solving for b. For example, using x = 1, y = 5, and m = 2 yields 5 = 2(1) + b. Solving for b yields b = 3. Therefore, the equation y = 2x + 3 could represent the relationship between x and y in the given table.

Alternatively, if an equation represents the relationship between x and y, then when each pair of x and y from the table is substituted into the equation, the result will be a true statement. Of the equations given, the equation y = 2x + 3 in choice A is the only equation that results in a true statement when each of the pairs of x and y are substituted into the equation.

Choices B, C, and D are incorrect because when at least one pair of x and y from the table is substituted into the equations given in these choices, the result is a false statement. For example, when the pair x = 4 and y = 11 is substituted into the equation in choice B, the result is 11 = 3(4) – 2, or 11 = 10, which is false.

Question 389 389 of 569 selected Linear Equations In 2 Variables M
 
Characteristics for Rock Types
Rock typeWeight per volume (lb/ft3)Cost per pound
Basalt180$0.18
Granite165$0.09
Limestone120$0.03
Sandstone135$0.22

A city is planning to build a rock retaining wall, a monument, and a garden in a park. The table above shows four rock types that will be considered for use in the project. Also shown for each rock type is its weight per volume, in pounds per cubic foot (lb/ft3), and the cost per pound, in dollars. Only basalt, granite, and limestone will be used in the garden. The rocks in the garden will have a total weight of 1,000 pounds. If 330 pounds of granite is used, which of the following equations could show the relationship between the amounts, x and y, in ft3, for each of the other rock types used?

 

  1. 165 x, plus, 180 y, equals 670

  2. 165 x, plus, 120 y, equals, 1,000

  3. 120 x, plus, 180 y, equals 670

  4. 120 x, plus, 180 y, equals, 1,000

Show Answer Correct Answer: C

Choice C is correct. It’s given that the weight of the granite used in the garden is 330 pounds. The weight of the limestone used in the garden is a product of its weight per volume, in lb/ft3, and its volume, in ft3. Therefore, the weight of the limestone used in the garden can be represented by 120 x, where x is the volume, in ft3, of the limestone used. Similarly, the weight of the basalt used in the garden can be represented by 180 y, where y is the volume, in ft3, of the basalt used. It’s given that the total weight of the rocks used in the garden will be 1,000 pounds. Thus, the sum of the weights of the three rock types used is 1,000 pounds, which can be represented by the equation 120 x, plus 180 y, plus 330, equals 1,000. Subtracting 330 from both sides of this equation yields 120 x, plus 180 y, equals 670.

Choice A is incorrect. This equation uses the weight per volume of granite instead of limestone. Choice B is incorrect. This equation uses the weight per volume of granite instead of basalt, and doesn’t take into account the 330 pounds of granite that will be used in the garden. Choice D is incorrect. This equation doesn’t take into account the 330 pounds of granite that will be used in the garden.

Question 390 390 of 569 selected Systems Of 2 Linear Equations In 2 Variables E

x = 8

x + 3 y = 26

The solution to the given system of equations is (x,y). What is the value of y ?

Show Answer Correct Answer: 6

The correct answer is 6 . The first equation in the given system is x = 8 . Substituting 8 for x in the second equation in the given system yields 8+3y=26. Subtracting 8 from both sides of this equation yields 3 y = 18 . Dividing both sides of this equation by 3 yields y = 6 . Therefore, the value of y is 6 .

Question 391 391 of 569 selected Linear Functions M

If f(x)=x+7 and g(x)=7x, what is the value of 4f(2)g(2)?

  1. -5

  2. 1

  3. 22

  4. 28

Show Answer Correct Answer: C

Choice C is correct. The value of f(2) can be found by substituting 2 for x in the given equation f(x)=x+7, which yields f(2)=2+7, or f(2)=9. The value of g(2) can be found by substituting 2 for x in the given equation g(x)=7x, which yields g(2)=7(2), or g(2)=14. The value of the expression 4f(2)-g(2) can be found by substituting the corresponding values into the expression, which gives 4(9)14. This expression is equivalent to 36-14, or 22 .

Choice A is incorrect. This is the value of f(2)-g(2), not 4f(2)-g(2).

Choice B is incorrect and may result from calculating 4f(2) as 4(2)+7, rather than
4(2+7).

Choice D is incorrect and may result from conceptual or calculation errors.

Question 392 392 of 569 selected Systems Of 2 Linear Equations In 2 Variables E
Equation 1: x plus y, equals 20.

Equation 2: 2 times, open parenthesis, x plus y, close parenthesis, plus 3 y, equals 85.

If the ordered pair x comma y is the solution to the given system of equations, what is the value of y ?

  1. 10

  2. 15

  3. 60

  4. 65

Show Answer Correct Answer: B

Choice B is correct. Substituting 20 for x plus y in the second equation in the system yields 2 times 20, plus 3y, equals 85, or 40 plus 3 y, equals 85. Subtracting 40 from both sides of this equation yields 3 y equals 45. Dividing both sides of this equation by 3 yields y equals 15.

Choice A is incorrect. If y equals 10, then x equals 10 since x plus y, equals 20. However, substituting 10 for both x and y in the second equation yields 70 equals 85, which is a false statement. Choice C is incorrect. If y equals 60, then x equals negative 40 since x plus y, equals 20. However, substituting these values for x and y in the second equation yields 220 equals 85, which is a false statement. Choice D is incorrect. If y equals 65, then x equals negative 45 since x plus y, equals 20. However, substituting these values for x and y in the second equation yields 235 equals 85, which is a false statement.

 

Question 393 393 of 569 selected Systems Of 2 Linear Equations In 2 Variables H

Equation 1: y equals, 2 x plus 1. Equation 2: y equals, a, x minus 8.

In the system of equations above, a is a constant. If the system of equations has no solution, what is the value of a ?

  1. negative one half

  2. 0

  3. 1

  4. 2

Show Answer Correct Answer: D

Choice D is correct. A system of two linear equations has no solution when the graphs of the equations have the same slope and different y-coordinates of the y-intercepts. Each of the given equations is written in the slope-intercept form of a linear equation, y equals, m x plus b, where m is the slope and b is the y-coordinate of the y-intercept of the graph of the equation. For these two linear equations, the y-coordinates of the y-intercepts are different: 1 and negative 8. Thus, if the system of equations has no solution, the slopes of the two linear equations must be the same. The slope of the first linear equation is 2. Therefore, for the system of equations to have no solution, the value of a must be 2.

Choices A, B, and C are incorrect and may result from conceptual and computational errors.

 

Question 394 394 of 569 selected Systems Of 2 Linear Equations In 2 Variables H

8 x + 7 y = 9
24 x + 21 y = 27

For each real number r , which of the following points lies on the graph of each equation in the xy-plane for the given system?

  1. (r,-8r7+97)

  2. (-8r7+97,r)

  3. (-8r7+9,8r7+27)

  4. (r3+9,-r3+27)

Show Answer Correct Answer: A

Choice A is correct. Dividing both sides of the second equation in the given system by 3 yields 8 x + 7 y = 9 , which is the first equation in the given system. Therefore, the first and second equations represent the same line in the xy-plane. If the x- and y-coordinates of a point satisfy an equation, the point lies on the graph of the equation in the xy-plane. Choice A is a point with x-coordinate r and y-coordinate - 8 r 7 + 9 7 . Substituting r for x and - 8 r 7 + 9 7 for y in the equation 8 x + 7 y = 9 yields 8r+7(-87r+97)=9. Applying the distributive property to the left-hand side of this equation yields 8r-8r+9=9. Combining like terms on the left-hand side of this equation yields 9=9, so the coordinates of the point (r,-87r+97) satisfy both equations in the given system. Therefore, for each real number r , the point (r,-87r+97) lies on the graph of each equation in the xy-plane for the given system.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 395 395 of 569 selected Linear Inequalities In 1 Or 2 Variables H

y<6x+2

For which of the following tables are all the values of x and their corresponding values of y solutions to the given inequality?

  1. x y
    3 20
    5 32
    7 44
  2. x y
    3 16
    5 36
    7 40
  3. x y
    3 16
    5 28
    7 40
  4. x y
    3 24
    5 36
    7 48
Show Answer Correct Answer: C

Choice C is correct. All the tables in the choices have the same three values of x , so each of the three values of x can be substituted in the given inequality to compare the corresponding values of y in each of the tables. Substituting 3 for x in the given inequality yields y<6(3)+2, or y<20. Therefore, when x = 3 , the corresponding value of y is less than 20 . Substituting 5 for x in the given inequality yields y<6(5)+2, or y<32. Therefore, when x = 5 , the corresponding value of y is less than 32 . Substituting 7 for x in the given inequality yields y<6(7)+2, or y<44. Therefore, when x = 7 , the corresponding value of y is less than 44 . For the table in choice C, when x = 3 , the corresponding value of y is 16 , which is less than 20 ; when x = 5 , the corresponding value of y is 28 , which is less than 32 ; when x = 7 , the corresponding value of y is 40 , which is less than 44 . Therefore, the table in choice C gives values of x and their corresponding values of y that are all solutions to the given inequality.

Choice A is incorrect. In the table for choice A, when x = 3 , the corresponding value of y is 20 , which is not less than 20 ; when x = 5 , the corresponding value of y is 32 , which is not less than 32 ; when x = 7 , the corresponding value of y is 44 , which is not less than 44 .

Choice B is incorrect. In the table for choice B, when x = 5 , the corresponding value of y is 36 , which is not less than 32 .

Choice D is incorrect. In the table for choice D, when x = 3 , the corresponding value of y is 24 , which is not less than 20 ; when x = 5 , the corresponding value of y is 36 , which is not less than 32 ; when x = 7 , the corresponding value of y is 48 , which is not less than 44 .

Question 396 396 of 569 selected Linear Equations In 2 Variables H

5 x + 7 y = 1

a x + b y = 1

In the given pair of equations, a and b are constants. The graph of this pair of equations in the xy-plane is a pair of perpendicular lines. Which of the following pairs of equations also represents a pair of perpendicular lines?

  1. 10 x + 7 y = 1

    a x - 2 b y = 1

  2. 10 x + 7 y = 1

    a x + 2 b y = 1

  3. 10 x + 7 y = 1

    2 a x + b y = 1

  4. 5 x - 7 y = 1

    a x + b y = 1

Show Answer Correct Answer: B

Choice B is correct. Two lines are perpendicular if their slopes are negative reciprocals, meaning that the slope of the first line is equal to -1 divided by the slope of the second line. Each equation in the given pair of equations can be written in slope-intercept form, y = m x + b , where m is the slope of the graph of the equation in the xy-plane and (0,b) is the y-intercept. For the first equation, 5 x + 7 y = 1 , subtracting 5 x from both sides gives 7y=-5x+1, and dividing both sides of this equation by 7 gives y=-57x+17. Therefore, the slope of the graph of this equation is - 5 7 . For the second equation, a x + b y = 1 , subtracting a x from both sides gives b y = - a x + 1 , and dividing both sides of this equation by b gives y=-abx+1b. Therefore, the slope of the graph of this equation is - a b . Since the graph of the given pair of equations is a pair of perpendicular lines, the slope of the graph of the second equation, - a b , must be the negative reciprocal of the slope of the graph of the first equation, - 5 7 . The negative reciprocal of - 5 7 is  -1(-57), or 7 5 . Therefore, - a b = 7 5 , or a b = - 7 5 . Similarly, rewriting the equations in choice B in slope-intercept form yields y=-107x+17 and y=-a2bx+12b. It follows that the slope of the graph of the first equation in choice B is - 10 7 and the slope of the graph of the second equation in choice B is - a 2 b . Since a b = - 7 5 , - a 2 b is equal to -(12)(-75), or 7 10 . Since 7 10 is the negative reciprocal of - 10 7 , the pair of equations in choice B represents a pair of perpendicular lines.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 397 397 of 569 selected Linear Functions E

On January 1, 2015, a city’s minimum hourly wage was $9.25. It will increase by $0.50 on the first day of the year for the next 5 years. Which of the following functions best models the minimum hourly wage, in dollars, x years after January 1, 2015, where x equals the following five values: 1, 2, 3, 4, 5 ?

  1. f of x equals, 9 point 2 5, minus 0 point 5 0 x

  2. f of x equals, 9 point 2 5 x, minus 0 point 5 0

  3. f of x equals, 9 point 2 5, plus 0 point 5 0 x

  4. f of x equals, 9 point 2 5 x, plus 0 point 5 0

Show Answer Correct Answer: C

Choice C is correct. It’s given that the city’s minimum hourly wage will increase by $0.50 on the first day of the year for the 5 years after January 1, 2015. Therefore, the total increase, in dollars, in the minimum hourly wage x years after January 1, 2015, is represented by 0 point 5 0 x. Since the minimum hourly wage on January 1, 2015, was $9.25, it follows that the minimum hourly wage, in dollars, x years after January 1, 2015, is represented by 9 point 2 5, plus 0 point 5 0 x. Therefore, the function f of x equals, 9 point 2 5, plus 0 point 5 0 x best models this situation.

Choices A, B, and D are incorrect. In choice A, the function models a situation where the minimum hourly wage is $9.25 on January 1, 2015, but decreases by $0.50 on the first day of the year for the next 5 years. The functions in choices B and D both model a situation where the minimum hourly wage is increasing by $9.25 on the first day of the year for the 5 years after January 1, 2015.

 

Question 398 398 of 569 selected Linear Equations In 1 Variable M

If 6 7 p+18 =54 , what is the value of 7 p ?

Show Answer Correct Answer: 294

The correct answer is 294 . Subtracting 18 from each side of the given equation yields 67p=36. Multiplying each side of this equation by 7 6 yields p = 42 . Multiplying each side of this equation by 7 yields 7 p = 294 . Therefore, the value of 7 p is 294 .

Question 399 399 of 569 selected Linear Inequalities In 1 Or 2 Variables H

A business owner plans to purchase the same model of chair for each of the 81 employees. The total budget to spend on these chairs is $14,000 , which includes a 7 % sales tax. Which of the following is closest to the maximum possible price per chair, before sales tax, the business owner could pay based on this budget?

  1. $148.15

  2. $161.53

  3. $172.84

  4. $184.94

Show Answer Correct Answer: B

Choice B is correct. It’s given that a business owner plans to purchase 81 chairs. If p is the price per chair, the total price of purchasing 81 chairs is 81 p . It’s also given that 7% sales tax is included, which is equivalent to 81 p multiplied by 1.07 , or 81(1.07)p. Since the total budget is $14,000, the inequality representing the situation is given by 81(1.07)p14,000. Dividing both sides of this inequality by 81(1.07) and rounding the result to two decimal places gives  p161.53. To not exceed the budget, the maximum possible price per chair is $161.53.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect. This is the maximum possible price per chair including sales tax, not the maximum possible price per chair before sales tax.

Choice D is incorrect. This is the maximum possible price if the sales tax is added to the total budget, not the maximum possible price per chair before sales tax.

Question 400 400 of 569 selected Linear Equations In 1 Variable E

What value of x is the solution to the equation 16 x + 24 = 24 x ?

  1. -4

  2. 3 10

  3. 1 3

  4. 3

Show Answer Correct Answer: D

Choice D is correct. Subtracting 16x from both sides of the given equation yields 24=8x. Dividing both sides of this equation by 8 yields 3=x. Therefore, the solution to the given equation is 3.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Question 401 401 of 569 selected Systems Of 2 Linear Equations In 2 Variables E

y=-3x

4x+y=15

The solution to the given system of equations is (x,y). What is the value of x ?

  1. 1

  2. 5

  3. 15

  4. 45

Show Answer Correct Answer: C

Choice C is correct. The given system of linear equations can be solved by the substitution method. Substituting -3x for y from the first equation in the given system into the second equation yields 4x+(-3x)=15, or x=15.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect. This is the absolute value of y , not the value of x .

Question 402 402 of 569 selected Linear Equations In 2 Variables H

5 G + 45 R = 380

At a school fair, students can win colored tokens that are worth a different number of points depending on the color. One student won G green tokens and R red tokens worth a total of 380 points. The given equation represents this situation. How many more points is a red token worth than a green token?

Show Answer Correct Answer: 40

The correct answer is 40 . It's given that 5G+45R=380, where G is the number of green tokens and R is the number of red tokens won by one student and these tokens are worth a total of 380 points. Since the equation represents the situation where the student won points with green tokens and red tokens for a total of 380 points, each term on the left-hand side of the equation represents the number of points won for one of the colors. Since the coefficient of G in the given equation is 5 , a green token must be worth 5 points. Similarly, since the coefficient of R in the given equation is 45 , a red token must be worth 45 points. Therefore, a red token is worth 45-5 points, or 40 points, more than a green token.

Question 403 403 of 569 selected Linear Inequalities In 1 Or 2 Variables H

In a set of four consecutive odd integers, where the integers are ordered from least to greatest, the first integer is represented by x . The product of 12 and the fourth odd integer is at most 26 less than the sum of the first and third odd integers. Which inequality represents this situation?

  1. 12(x+6)x+(x+4)-26

  2. 12(x+6)26-(x+(x+4))

  3. 12(x+4)x+(x+3)-26

  4. 12(x+4)26-(x+(x+3))

Show Answer Correct Answer: A

Choice A is correct. It’s given that the four odd integers are consecutive, ordered from least to greatest, and that the first odd integer is represented by x . It follows that the second odd integer is represented by x + 2 , the third odd integer is represented by x + 4 , and the fourth odd integer is represented by x + 6 . Therefore, the product of 12 and the fourth odd integer is represented by 12(x+6), and 26 less than the sum of the first and third odd integers is represented by x+(x+4)-26. Since the product of 12 and the fourth odd integer is at most 26 less than the sum of the first and third odd integers, it follows that 12(x+6)x+(x+4)-26.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 404 404 of 569 selected Linear Functions E

A student council group is selling school posters for a fundraiser. They use the function p(x)=5x-220 to determine their profit p(x), in dollars, for selling x school posters. In order to earn a profit of $900, how many school posters must they sell?

Show Answer Correct Answer: 224

The correct answer is 224 . It’s given that a student council group uses the function p(x)=5x-220 to determine their profit p(x), in dollars, for selling x school posters. Substituting 900 for p(x) in the given function yields 900 = 5 x - 220 . Adding 220 to each side of this equation yields 1,120 = 5 x . Dividing each side of this equation by 5 yields 224 = x . Therefore, in order to earn a profit of $900, they must sell 224 school posters.

Question 405 405 of 569 selected Systems Of 2 Linear Equations In 2 Variables E

4 x - 3 y = 5

x = 8

What is the solution (x,y) to the given system of equations?

  1. (8,9)

  2. (8,-24)

  3. (8,-9)

  4. (8,24)

Show Answer Correct Answer: A

Choice A is correct. The second equation in the given system is x=8. Substituting 8 for x in the first equation in the given system yields 4(8)-3y=5, or 32-3y=5. Subtracting 32 from both sides of this equation yields -3y=-27. Dividing both sides of this equation by -3 yields y=9. Therefore, the solution (x,y) to the given system of equations is (8,9).

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 406 406 of 569 selected Linear Functions E

Argon is placed inside a container with a constant volume. The graph shows the estimated pressure y , in pounds per square inch (psi), of the argon when its temperature is x kelvins.

  • The line slants gradually up from left to right.
  • The line begins at the approximate point (90 comma 1.8).
  • The line passes through the following points:
    • (300 comma 6)
    • (450 comma 9)
    • (600 comma 12)

What is the estimated pressure of the argon, in psi, when the temperature is 600 kelvins?

  1. 6

  2. 12

  3. 300

  4. 600

Show Answer Correct Answer: B

Choice B is correct. For the graph shown, the x-axis represents temperature, in kelvins, and the y-axis represents the estimated pressure, in pounds per square inch (psi). The estimated pressure of the argon when the temperature is 600 kelvins can be found by locating the point on the graph where the value of x is equal to 600 . The graph passes through the point (600,12). This means that when the temperature is 600 kelvins, the estimated pressure is 12 psi.

Choice A is incorrect. This is the estimated pressure, in psi, of the argon when the temperature is 300 kelvins, not 600 kelvins.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect. This is the temperature, in kelvins, of the argon.

Question 407 407 of 569 selected Linear Functions E

The length, y , of a white whale was 162 centimeters (cm) when it was born and increased an average of 4.8 cm per month for the first 12 months after it was born. Which equation best represents this situation, where x is the number of months after the whale was born and y is the length, in cm, of the whale?

  1. y = 162 x

  2. y = 162 x + 162

  3. y = 4.8 x + 4.8

  4. y = 4.8 x + 162

Show Answer Correct Answer: D

Choice D is correct. It's given that the length of the whale was 162 cm when it was born and that its length increased an average of 4.8 cm per month for the first 12 months after it was born. Since x represents the number of months after the whale was born, the total increase in the whale's length, in cm, is 4.8 times x , or 4.8 x . The length of the whale y , in cm, can be found by adding the whale's length at birth, 162 cm, to the total increase in length, 4.8x cm. Therefore, the equation that best represents this situation is y = 4.8 x + 162 .

Choice A is incorrect and may result from conceptual errors.

Choice B is incorrect and may result from conceptual errors.

Choice C is incorrect and may result from conceptual errors.

Question 408 408 of 569 selected Linear Inequalities In 1 Or 2 Variables E

  • The boundary of the inequality is a solid line.
    • The line slants gradually up from left to right.
    • The line passes through the following points:
      • (0 comma negative 6)
      • (9 comma 0)
  • The area above and to the left of the boundary is shaded.

The shaded region shown represents the solutions to which inequality?

  1. y23x-6

  2. y23x+6

  3. y23x-9

  4. y23x+9

Show Answer Correct Answer: A

Choice A is correct. The equation for the line representing the boundary of the shaded region can be written in slope-intercept form y = m x + b , where m is the slope and (0,b) is the y-intercept of the line. For the graph shown, the boundary line passes through the points (0,-6) and (9,0). Given two points on a line, (x1,y1) and (x2,y2), the slope of the line can be calculated using the equation m=y2-y1x2-x1. Substituting the points (0,-6) and (9,0) for (x1,y1) and (x2,y2), respectively, in this equation yields m=0-(-6)9-0, which is equivalent to m=69, or m = 2 3 . Since the point (0,-6) represents the y-intercept, it follows that b = - 6 . Substituting 2 3 for m and - 6 for b in the equation y = m x + b yields y=23x-6 as the equation of the boundary line. Since the shaded region represents all the points on and above this boundary line, it follows that the shaded region shown represents the solutions to the inequality y23x-6.

Choice B is incorrect. This inequality represents a region whose boundary line has a y-intercept of (0,6), not (0,-6).

Choice C is incorrect. This inequality represents a region whose boundary line has a y-intercept of (0,-9), not (0,-6).

Choice D is incorrect. This inequality represents a region whose boundary line has a y-intercept of (0,9), not (0,-6).

Question 409 409 of 569 selected Systems Of 2 Linear Equations In 2 Variables H

Which system of linear equations has no solution?

  1. - 2 x + 3 y = -9

    2 x - 3 y = 9

  2. 2 x - 3 y = 9

    3 x + 4 y = 10

  3. 2 x - 3 y = 9

    - 6 x + 9 y = -27

  4. - 2 x + 3 y = 9

    4 x - 6 y = 18

Show Answer Correct Answer: D

Choice D is correct. A system of linear equations can be solved by the elimination method. Multiplying the equation -2x+3y=9 by 2 yields -4x+6y=18. Adding this equation to the equation 4x-6y=18 yields 0=36, which has no solution. It follows that the system of linear equations consisting of -2x+3y=9 and 4x-6y=18 has no solution.

Choice A is incorrect. This system of linear equations has infinitely many solutions, rather than no solution.

Choice B is incorrect. This system of linear equations has one solution, rather than no solution.

Choice C is incorrect. This system of linear equations has infinitely
many solutions, rather than no solution.

Question 410 410 of 569 selected Systems Of 2 Linear Equations In 2 Variables H

Store A sells raspberries for $5.50  per pint and blackberries for $3.00 per pint. Store B sells raspberries for $6.50 per pint and blackberries for $8.00 per pint. A certain purchase of raspberries and blackberries would cost $37.00 at Store A or $66.00 at Store B. How many pints of blackberries are in this purchase?

  1. 4

  2. 5

  3. 8

  4. 12

Show Answer Correct Answer: B

Choice C is correct. It’s given that store A sells raspberries for $5.50 per pint and blackberries for $3.00 per pint, and a certain purchase of raspberries and blackberries at store A would cost $37.00. It’s also given that store B sells raspberries for $6.50 per pint and blackberries for $8.00 per pint, and this purchase of raspberries and blackberries at store B would cost $66.00. Let r represent the number of pints of raspberries and b represent the number of pints of blackberries in this purchase. The equation 5.50r+3.00b=37.00 represents this purchase of raspberries and blackberries from store A and the equation 6.50r+8.00b=66.00 represents this purchase of raspberries and blackberries from store B. Solving the system of equations by elimination gives the value of r and the value of b that make the system of equations true. Multiplying both sides of the equation for store A by 6.5 yields (5.50r)(6.5)+(3.00b)(6.5)=(37.00)(6.5), or 35.75r+19.5b=240.5. Multiplying both sides of the equation for store B by 5.5 yields (6.50r)(5.5)+(8.00b)(5.5)=(66.00)(5.5), or 35.75r+44b=363. Subtracting both sides of the equation for store A, 35.75r+19.5b=240.5, from the corresponding sides of the equation for store B, 35.75r+44b=363, yields (35.75r-35.75r)+(44b-19.5b)=(363-240.5), or 24.5b=122.5. Dividing both sides of this equationby 24.5 yields b=5. Thus, 5 pints of blackberries are in
this purchase.


Choices A and B are incorrect and may result from conceptual or calculation errors. Choice D is incorrect. This is the number of pints of raspberries, not blackberries, in the purchase.

Question 411 411 of 569 selected Linear Equations In 2 Variables M

Line r is defined by the equation 4 x - 9 y = 3 . Line s is parallel to line r in the xy-plane. What is the slope of line s ?

  1. 9 4

  2. 4 9

  3. -4

  4. -9

Show Answer Correct Answer: B

Choice B is correct. It's given that line s is parallel to line r in the xy-plane. This means that the slope of line r is equal to the slope of line s. Line r is defined by the equation 4x-9y=3. This equation can be written in slope-intercept form y=mx+b, where m represents the slope of the line and b represents the y-coordinate of the y-intercept of the line. Subtracting 4x from both sides of the equation 4x-9y=3 yields -9y=-4x+3. Dividing both sides of this equation by -9 yields y=49x-13. Therefore, the slope of line r is 49. Since parallel lines have equal slopes, the slope of line s is also 49.

Choice A is incorrect. This is the reciprocal of the slope of line s, not the slope of line s.

Choice C is incorrect and may result from conceptual or calculation errors. 

Choice D is incorrect and may result from conceptual or calculation errors. 

Question 412 412 of 569 selected Linear Equations In 2 Variables E

The equation x+y=1,440 represents the number of minutes of daylight (between sunrise and sunset), x , and the number of minutes of non-daylight, y , on a particular day in Oak Park, Illinois. If this day has 670 minutes of daylight, how many minutes of non-daylight does it have?

  1. 670

  2. 770

  3. 1,373

  4. 1,440

Show Answer Correct Answer: B

Choice B is correct. It’s given that the equation x+y=1,440 represents the number of minutes of daylight, x , and the number of minutes of non-daylight, y , on a particular day in Oak Park, Illinois. It’s also given that this day has 670 minutes of daylight. Substituting 670 for x in the equation x+y=1,440 yields 670+y=1,440. Subtracting 670 from both sides of this equation yields y=770. Therefore, this day has 770 minutes of non-daylight.

Choice A is incorrect. This is the number of minutes of daylight, not non-daylight, on this day.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect. This is the total number of minutes of daylight and non-daylight.

Question 413 413 of 569 selected Linear Functions H
xnegative 11negative 10negative 9negative 8
f(x)21181512

 

The table above shows some values of x and their corresponding values f of x for the linear function f. What is the x-intercept of the graph of y equals f of x in the xy-plane?

  1. the point with coordinates negative 3 comma 0

  2. the point with coordinates negative 4 comma 0

  3. the point with coordinates negative 9 comma 0

  4. the point with coordinates negative 12 comma 0

Show Answer Correct Answer: B

Choice B is correct. The equation of a linear function can be written in the form y equals, m x plus b, where y equals f of x, m is the slope of the graph of y equals f of x, and b is the y-coordinate of the y-intercept of the graph. The value of m can be found using the slope formula, m equals the fraction with numerator y sub 2 minus y sub 1, and denominator x sub 2 minus x sub 1, end fraction. According to the table, the points with coordinates negative 11 comma 21 and with coordinates negative 10 comma 18 lie on the graph of y equals f of x. Using these two points in the slope formula yieldsm equals the fraction with numerator 18 minus 21, and denominator negative 10 plus 11, end fraction, or negative 3. Substituting negative 3 for m in the slope-intercept form of the equation yields y equals, negative 3 x plus b. The value of b can be found by substituting values from the table and solving; for example, substituting the coordinates of the point with coordinates negative 11 comma 21 into the equation y equals, negative 3 x plus b gives 21 equals, negative 3 times negative 11, plus b, which yields b equals negative 12. This means the function given by the table can be represented by the equation y equals, negative 3 x minus 12. The value of the x-intercept of the graph of y equals f of x can be determined by finding the value of x when y equals 0. Substituting y equals 0 into y equals, negative 3 x minus 12 yields 0 equals, negative 3 x minus 12, or x equals negative 4. This corresponds to the point with coordinates negative 4 comma 0.

Choice A is incorrect and may result from substituting the value of the slope for the x-coordinate of the x-intercept. Choice C is incorrect and may result from a calculation error. Choice D is incorrect and may result from substituting the y-coordinate of the y-intercept for the x-coordinate of the x-intercept.

 

Question 414 414 of 569 selected Linear Equations In 1 Variable E

If one half x, minus one sixth x, equals 1, what is the value of x ?

  1. negative 4

  2. one third

  3. 3

  4. 6

Show Answer Correct Answer: C

Choice C is correct. To make it easier to add like terms on the left-hand side of the given equation, both sides of the equation can be multiplied by 6, which is the lowest common denominator of one half and one sixth. This yields 3 x minus x, equals 6, which can be rewritten as 2 x equals 6. Dividing both sides of this equation by 2 yields x equals 3.

Choice A is incorrect and may result from subtracting the denominators instead of numerators with common denominators to get negative one fourth x, rather than one third x, on the left-hand side of the equation. Choice B is incorrect and may result from rewriting the given equation as one half x, equals one sixth instead of 2 x equals 6. Choice D is incorrect and may result from conceptual or computational errors.

 

Question 415 415 of 569 selected Linear Equations In 2 Variables M

What is the slope of the graph of y = 5 x 13 - 23 in the xy-plane?

Show Answer Correct Answer: .3846, 5/13

The correct answer is 5 13 . The graph of a line in the xy-plane can be represented by the equation y = m x + b , where m is the slope of the line and b is the y-coordinate of the y-intercept. The given equation can be written as y=(513)x-23. Therefore, the slope of the graph of this equation in the xy-plane is 5 13 . Note that 5/13, .3846, 0.385, and 0.384 are examples of ways to enter a correct answer.

Question 416 416 of 569 selected Systems Of 2 Linear Equations In 2 Variables M

Which of the following systems of linear equations has no solution?

  1. y = 6 x + 3

    y = 6 x + 9

  2. y = 10

    y = 10 x + 10

  3. y = 14 x + 14

    y = 10 x + 14

  4. x = 3

    y = 10

Show Answer Correct Answer: A

Choice A is correct. A system of two linear equations in two variables, x and y , has no solution if the graphs of the lines represented by the equations in the xy-plane are distinct and parallel. The graphs of two lines in the xy-plane represented by equations in slope-intercept form, y=mx+b, where m and b are constants, are parallel if their slopes, m , are the same and are distinct if their y-coordinates of the y-intercepts, b , are different. In the equations y=6x+3 and y=6x+9, the values of m are each 6 , and the values of b are 3 and 9 , respectively. Since the slopes of these lines are the same and the y-coordinates of the y-intercepts are different, it follows that the system of linear equations in choice A has no solution.

Choice B is incorrect. The two lines represented by these equations are a horizontal line and a line with a slope of 10 that have the same y-coordinate of the y-intercept. Therefore, this system has a solution, (0,10), rather than no solution.

Choice C is incorrect. The two lines represented by these equations have different slopes and the same y-coordinate of the y-intercept. Therefore, this system has a solution, (0,14), rather than no solution.

Choice D is incorrect. The two lines represented by these equations are a vertical line and a horizontal line. Therefore, this system has a solution, (3,10), rather than no solution.

Question 417 417 of 569 selected Linear Functions M
x g(x)
1 54
2 51
3 48
4 45

For the linear function g , the table shows four values of x and their corresponding values of g(x). The function can be written as g(x)=mx+b, where m and b are constants. What is the value of b ?

  1. 3

  2. 27

  3. 54

  4. 57

Show Answer Correct Answer: D

Choice D is correct. It's given that for the linear function g, the table shows four values of x and their corresponding values of g(x). It's also given that the function can be written as g(x)=mx+b, where m and b are constants. The table shows that when the value of x is 1, the corresponding value of g(x) is 54. Substituting 1 for x and 54 for g(x) in g(x)=mx+b yields 54=m(1)+b or, 54=m+b. Subtracting b from both sides of this equation yields 54-b=m. The table also shows that when the value of x is 2, the corresponding value of  g(x) is 51. Substituting 2 for x and 51 for g(x) in g(x)=mx+b yields 51=m(2)+b, or 51=2m+b. Substituting 54-b for m in this equation yields 51=2(54-b)+b. Applying the distributive property to the right-hand side of this equation yields 51=108-2b+b, or 51=108-b. Subtracting 108 from both sides of this equation yields -57=-b. Dividing both sides of this equation by -1 yields 57=b.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Question 418 418 of 569 selected Linear Functions E

The function f is defined by f(x)=110x-2. What is the y-intercept of the graph of y=f(x) in the xy-plane?

  1. (-2,0)

  2. (0,-2)

  3. (0,110)

  4. (110,0)

Show Answer Correct Answer: B

Choice B is correct. The y-intercept of the graph of a function in the xy-plane is the point on the graph where x=0. It′s given that f(x)=110x-2. Substituting 0 for x in this equation yields f(0)=110(0)-2, or f(0)=-2. Since it′s given that y=f(x), it follows that y=-2 when x=0. Therefore, the y-intercept of the graph of y=f(x) in the xy-plane is (0,-2).

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 419 419 of 569 selected Linear Functions M

g(m)=-0.05m+12.1

The given function g models the number of gallons of gasoline that remains from a full gas tank in a car after driving m miles. According to the model, about how many gallons of gasoline are used to drive each mile?

  1. 0.05

  2. 12.1

  3. 20

  4. 242.0

Show Answer Correct Answer: A

Choice A is correct. It's given that the function g models the number of gallons that remain from a full gas tank in a car after driving m miles. In the given function g(m)=-0.05m+12.1, the coefficient of m is -0.05 . This means that for every increase in the value of m by 1 , the value of g(m) decreases by 0.05 . It follows that for each mile driven, there is a decrease of 0.05 gallons of gasoline. Therefore, 0.05 gallons of gasoline are used to drive each mile.

Choice B is incorrect and represents the number of gallons of gasoline in a full gas tank.

Choice C is incorrect and may result from conceptual errors.

Choice D is incorrect and may result from conceptual errors.

Question 420 420 of 569 selected Linear Inequalities In 1 Or 2 Variables H

11x+14y115

Anthony will spend at most $115 to purchase x small cheese pizzas and y large cheese pizzas for a team dinner. The given inequality represents this situation. Which of the following is the best interpretation of 14 y in this context?

  1. The amount, in dollars, Anthony will spend on each large cheese pizza

  2. The amount, in dollars, Anthony will spend on each small cheese pizza

  3. The total amount, in dollars, Anthony will spend on large cheese pizzas

  4. The total amount, in dollars, Anthony will spend on small cheese pizzas

Show Answer Correct Answer: C

Choice C is correct. It's given that Anthony will spend at most $115 to purchase x small cheese pizzas and y large cheese pizzas. In the inequality 11x+14y115, y represents the number of large cheese pizzas that Anthony will purchase. This means the coefficient 14 represents the amount, in dollars, Anthony will spend on each large cheese pizza. Therefore, the best interpretation of 14y in this context is the total amount, in dollars, Anthony will spend on large cheese pizzas.

Choice A is incorrect. This is the best interpretation of 14, not 14y.

Choice B is incorrect. This is the best interpretation of 11, not 14y.

Choice D is incorrect. This is the best interpretation of 11x, not 14y.

Question 421 421 of 569 selected Systems Of 2 Linear Equations In 2 Variables M

Which of the following systems of linear equations has no solution?

  1. x = 3

    y = 5

  2. y = 6 x + 6

    y = 5 x + 6

  3. y = 16 x + 3

    y = 16 x + 19

  4. y = 5

    y = 5 x + 5

Show Answer Correct Answer: C

Choice C is correct. A system of two linear equations in two variables, x and y , has no solution if the graphs of the lines represented by the equations in the xy-plane are distinct and parallel. The graphs of two lines in the xy-plane represented by equations in slope-intercept form, y=mx+b, where m and b are constants, are parallel if their slopes, m , are the same and are distinct if their y-coordinates of the y-intercepts, b , are different. In the equations y=16x+3 and y=16x+19, the values of m are each 16 , and the values of b are 3 and 19 , respectively. Since the slopes of these lines are the same, and the y-coordinates of the y-intercepts are different, it follows that the system of linear equations in choice C has no solution. 

Choice A is incorrect. The lines represented by the equations in this system are a vertical line and a horizontal line. Therefore, this system has a solution, (3,5), rather than no solution.

Choice B is incorrect. The two lines represented by these equations have different slopes and the same y-coordinate of the y-intercept. Therefore, this system has a solution, (0,6), rather than no solution.

Choice D is incorrect. The two lines represented by these equations are a horizontal line and a line with a slope of 5 that have the same y-coordinate of the y-intercept. Therefore, this system has a solution, (0,5), rather than no solution. 

Question 422 422 of 569 selected Linear Equations In 2 Variables E

Line k is defined by y=14x+1. Line j is parallel to line k in the xy-plane. What is the slope of j ?

Show Answer Correct Answer: .25, 1/4

The correct answer is 14. It's given that line k is defined by y=14x+1. It's also given that line j is parallel to line k in the xy-plane. A line in the xy-plane represented by an equation in slope-intercept form y=mx+b has a slope of m and a y-intercept of (0,b). Therefore, the slope of line k is 14. Since parallel lines have equal slopes, the slope of line j is 14. Note that 1/4 and .25 are examples of ways to enter a correct answer.

Question 423 423 of 569 selected Linear Equations In 1 Variable E

3 x +5 (x+4)=76

What value of x is the solution to the given equation?

  1. 7

  2. 8

  3. 56

  4. 72

Show Answer Correct Answer: A

Choice A is correct. Applying the distributive property on the left-hand side of the given equation yields 3x+5x+20=76, or 8 x + 20 = 76 . Subtracting 20 from each side of this equation yields 8 x = 56 . Dividing each side of this equation by 8 yields x = 7 .

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect. This is the solution to the equation x + 4 = 76 , not 3x+5(x+4)=76.

Question 424 424 of 569 selected Linear Inequalities In 1 Or 2 Variables H

Ken is working this summer as part of a crew on a farm. He earned $8 per hour for the first 10 hours he worked this week. Because of his performance, his crew leader raised his salary to $10 per hour for the rest of the week. Ken saves 90% of his earnings from each week. What is the least number of hours he must work the rest of the week to save at least $270 for the week?

  1. 38

  2. 33

  3. 22

  4. 16

Show Answer Correct Answer: C

Choice C is correct. Ken earned $8 per hour for the first 10 hours he worked, so he earned a total of $80 for the first 10 hours he worked. For the rest of the week, Ken was paid at the rate of $10 per hour. Let x be the number of hours he will work for the rest of the week. The total of Ken’s earnings, in dollars, for the week will be 10 x plus 80. He saves 90% of his earnings each week, so this week he will save 0 point 9 times, open parenthesis, 10 x plus 80, close parenthesis dollars. The inequality 0 point 9 times, open parenthesis, 10 x plus 80, close parenthesis, is greater than or equal to 270 represents the condition that he will save at least $270 for the week. Factoring 10 out of the expression 10 x plus 80 gives 10 times, open parenthesis, x plus 8, close parenthesis. The product of 10 and 0.9 is 9, so the inequality can be rewritten as 9 times, open parenthesis, x plus 8, close parenthesis, is greater than or equal to 270. Dividing both sides of this inequality by 9 yields x plus 8, is greater than or equal to 30, so x is greater than or equal to 22. Therefore, the least number of hours Ken must work the rest of the week to save at least $270 for the week is 22.

Choices A and B are incorrect because Ken can save $270 by working fewer hours than 38 or 33 for the rest of the week. Choice D is incorrect. If Ken worked 16 hours for the rest of the week, his total earnings for the week will be 80 dollars plus 160 dollars, equals 240 dollars, which is less than $270. Since he saves only 90% of his earnings each week, he would save even less than $240 for the week.

Question 425 425 of 569 selected Systems Of 2 Linear Equations In 2 Variables E

  • For the first line in the system:
    • The line slants sharply down from left to right.
    • The line passes through the following points:
      • (4 comma 1)
      • (StartFraction 32 Over 7 EndFraction comma 0)
  • For the second line in the system:
    • The line slants sharply up from left to right.
    • The line passes through the following points:
      • (StartFraction 32 Over 9 EndFraction comma 0)
      • (4 comma 1)

The graph of a system of linear equations is shown. The solution to the system is (x,y). What is the value of x ?

Show Answer Correct Answer: 4

The correct answer is 4 . A solution to a system of equations must satisfy each equation in the system. It follows that if (x,y) is a solution to the system, the point (x,y) lies on the graph in the xy-plane of each equation in the system. According to the graph, the point (x,y) that lies on the graph of each equation in the system is (4,1). Therefore, the solution to the system is (4,1). It follows that the value of x is 4 .

Question 426 426 of 569 selected Linear Inequalities In 1 Or 2 Variables H
 
 y is greater than 2 x minus 1, and, 2 x is greater than 5
 

Which of the following consists of the y-coordinates of all the points that satisfy the system of inequalities above?

  1. y is greater than 6

  2. y is greater than 4

  3. y is greater than five-halves

  4. y is greater than three-halves

Show Answer Correct Answer: B

Choice B is correct. Subtracting the same number from each side of an inequality gives an equivalent inequality. Hence, subtracting 1 from each side of the inequality 2 x is greater than 5 gives 2 x minus 1, is greater than 4. So the given system of inequalities is equivalent to the system of inequalities y is greater than, 2 x minus 1 and 2 x minus 1, is greater than 4, which can be rewritten as y is greater than, 2 x minus 1, which is greater than 4. Using the transitive property of inequalities, it follows that y is greater than 4.

Choice A is incorrect because there are points with a y-coordinate less than 6 that satisfy the given system of inequalities. For example, the point with coordinates 3 comma 5 point 5 satisfies both inequalities. Choice C is incorrect. This may result from solving the inequality 2 x is greater than 5 for x, then replacing x with y. Choice D is incorrect because this inequality allows y-values that are not the y-coordinate of any point that satisfies both inequalities. For example, y equals 2 is contained in the set y is greater than three halves; however, if 2 is substituted into the first inequality for y, the result is x is less than three halves. This cannot be true because the second inequality gives x is greater than five halves.

 

Question 427 427 of 569 selected Linear Functions H

Kaylani used fabric measuring 5 yards in length to make each suit for a men's choir. The relationship between the number of suits that Kaylani made, x , and the total length of fabric that she purchased y , in yards, is represented by the equation y-5x=6. What is the best interpretation of 6 in this context?

  1. Kaylani made 6 suits.

  2. Kaylani purchased a total of 6 yards of fabric.

  3. Kaylani used a total of 6 yards of fabric to make the suits.

  4. Kaylani purchased 6 yards more fabric than she used to make the suits.

Show Answer Correct Answer: D

Choice D is correct. It’s given that the equation y-5x=6 represents the relationship between the number of suits that Kaylani made, x , and the total length of fabric she purchased, y , in yards. Adding 5 x to both sides of the given equation yields y = 5 x + 6 . Since Kaylani made x suits and used 5 yards of fabric to make each suit, the expression 5 x represents the total amount of fabric she used to make the suits. Since y represents the total length of fabric Kaylani purchased, in yards, it follows from the equation y = 5 x + 6 that Kaylani purchased 5 x yards of fabric to make the suits, plus an additional 6 yards of fabric. Therefore, the best interpretation of 6 in this context is that Kaylani purchased 6 yards more fabric than she used to make the suits.

Choice A is incorrect. Kaylani made a total of x suits, not 6 suits.

Choice B is incorrect. Kaylani purchased a total of y yards of fabric, not a total of 6 yards of fabric.

Choice C is incorrect. Kaylani used a total of 5 x yards of fabric to make the suits, not a total of 6 yards of fabric.

Question 428 428 of 569 selected Linear Equations In 2 Variables E

For a camping trip a group bought x one-liter bottles of water and y three-liter bottles of water, for a total of 240 liters of water. Which equation represents this situation?

  1. x+3y=240

  2. x+y=240

  3. 3x+3y=240

  4. 3x+y=240

Show Answer Correct Answer: A

Choice A is correct. It's given that for a camping trip a group bought x one-liter bottles of water and y three-liter bottles of water. Since the group bought x one-liter bottles of water, the total number of liters bought from x one-liter bottles of water is represented as 1x, or x . Since the group bought y three-liter bottles of water, the total number of liters bought from y three-liter bottles of water is represented as 3 y . It's given that the group bought a total of 240 liters; thus, the equation x + 3 y = 240 represents this situation.

Choice B is incorrect and may result from conceptual errors.

Choice C is incorrect and may result from conceptual errors.

Choice D is incorrect. This equation represents a situation where the group bought x three-liter bottles of water and y one-liter bottles of water, for a total of 240 liters of water.

Question 429 429 of 569 selected Linear Equations In 2 Variables H
The figure presents the graph of a line in a coordinate plane, with the origin labeled O. The horizontal axis is labeled “Number of hours at job A,” and the numbers 5 through 20, in increments of 5, are indicated. The vertical axis is labeled “Number of hours at job B,” and the numbers 5 and 10 are indicated. The line begins on the vertical axis at 8, and slants downward and to the right until it ends on the horizontal axis at 16.

To earn money for college, Avery works two part-time jobs: A and B. She earns $10 per hour working at job A and $20 per hour working at job B. In one week, Avery earned a total of s dollars for working at the two part-time jobs. The graph above represents all possible combinations of numbers of hours Avery could have worked at the two jobs to earn s dollars. What is the value of s ?

  1. 128

  2. 160

  3. 200

  4. 320

Show Answer Correct Answer: B

Choice B is correct. Avery earns $10 per hour working at job A. Therefore, if she works a hours at job A, she will earn 10 a dollars. Avery earns $20 per hour working at job B. Therefore, if she works b hours at job B, she will earn 20 b dollars. The graph shown represents all possible combinations of the number of hours Avery could have worked at the two jobs to earn s dollars. Therefore, if she worked a hours at job A, worked b hours at job B, and earned s dollars from both jobs, the following equation represents the graph: 10 a, plus 20 b, equals s, where s is a constant. Identifying any point with coordinates a, comma b from the graph and substituting the values of the coordinates for a and b, respectively, in this equation yield the value of s. For example, the point with coordinates 16 comma 0 , where a, equals 16 and b equals 0, lies on the graph. Substituting 16 for a and 0 for b in the equation 10 a, plus 20 b, equals s yields 10 times 16, plus, 20 times 0, equals s, or 160 equals s. Similarly, the point with coordinates 0 comma 8, where a, equals 0 and b equals 8, lies on the graph. Substituting 0 for a and 8 for b in the equation 10 a, plus 20 b, equals s yields 10 times 0, plus, 20 times 8, equals s, or 160 equals s.

Choices A, C, and D are incorrect. If the value of s is 128, 200, or 320, then no points with coordinates a, comma b on the graph will satisfy this equation. For example, if the value of s is 128 (choice A), then the equation 10 a, plus 20 b, equals s becomes 10 a, plus 20 b, equals 128. The point the point with coordinates 16 comma 0 , where a, equals 16 and b equals 0, lies on the graph. However, substituting 16 for a and 0 for b in 10 a, plus 20 b, equals s yields 10 times 16, plus, 20 times 0, equals 128, or 160 equals 128, which is false. Therefore, the point with coordinates 16 comma 0 doesn’t satisfy the equation, and so the value of s can’t be 128. Similarly, if s equals 200 (choice C) or s equals 320 (choice D), then substituting 16 for a and 0 for b yields 160 equals 200 and 160 equals 320, respectively, which are both false.

Question 430 430 of 569 selected Systems Of 2 Linear Equations In 2 Variables H

-x-wy=-337

2x-wy=47

In the given system of equations, w is a constant. In the xy-plane, the graphs of these equations intersect at the point (q,19), where q is a constant. What is the value of w ?

Show Answer Correct Answer: 11

The correct answer is 11. It’s given that the graphs of the equations in the given system intersect at the point (q,19), where q is a constant. Therefore, the coordinates of this point must satisfy both equations. Substituting the point (q,19) into the first equation, -x-wy=-337, yields -q-w(19)=-337. Adding 19w to both sides of this equation yields -q=-337+19w, which is equivalent to q=337-19w. Substituting the point (q,19) into the second equation yields 2q-w(19)=47. Substituting 337-19w in place of q in the equation 2q-w(19)=47 yields 2(337-19w)-19w=47. Applying the distributive property to the left-hand side of this equation yields 67438w19w=47. Combining like terms on the left-hand side of this equation yields 67457w=47. Subtracting 674 from both sides of this equation yields 57w=627. Dividing both sides of this equation by 57 yields w=11.

Question 431 431 of 569 selected Linear Inequalities In 1 Or 2 Variables E

For a 3 -week period in a town in Illinois, the lowest recorded temperature was 31 degrees Fahrenheit (°F) and the highest recorded temperature was 67°F. Which inequality is true for any recorded temperature t , in °F, in this town for this 3 -week period?

  1. t98

  2. t67

  3. 31t67

  4. t31

Show Answer Correct Answer: C

Choice C is correct. It's given that for a 3-week period in a town in Illinois, the lowest recorded temperature was 31°F and the highest recorded temperature was 67°F. It follows that the inequality 31t67 is true for any recorded temperature t, in °F, in this town for this 3-week period.

Choice A is incorrect and may result from conceptual errors.

Choice B is incorrect and may result from conceptual errors.

Choice D is incorrect and may result from conceptual errors.

Question 432 432 of 569 selected Linear Equations In 1 Variable E

If 2+x=60, what is the value of 16+8x?

Show Answer Correct Answer: 480

The correct answer is 480 . Multiplying both sides of the given equation by 8 yields 8(2+x)=8(60), or 16+8x=480. Therefore, if 2+x=60, the value of 16+8x is 480 .

Question 433 433 of 569 selected Linear Functions M

y equals, 18, minus 5 x

The equation above represents the speed y, in feet per second, of Sheila’s bicycle x seconds after she applied the brakes at the end of a ride. If the equation is graphed in the xy-plane, which of the following is the best interpretation of the x-coordinate of the line’s x-intercept in the context of the problem?

  1. The speed of Sheila’s bicycle, in feet per second, before Sheila applied the brakes

  2. The number of feet per second the speed of Sheila’s bicycle decreased each second after Sheila applied the brakes

  3. The number of seconds it took from the time Sheila began applying the brakes until the bicycle came to a complete stop

  4. The number of feet Sheila’s bicycle traveled from the time she began applying the brakes until the bicycle came to a complete stop

Show Answer Correct Answer: C

Choice C is correct. It’s given that for each point with coordinates x comma y on the graph of the given equation, the x-coordinate represents the number of seconds after Sheila applied the brakes, and the y-coordinate represents the speed of Sheila’s bicycle at that moment in time. For the graph of the equation, the y-coordinate of the x-intercept is 0. Therefore, the x-coordinate of the x-intercept of the graph of the given equation represents the number of seconds it took from the time Sheila began applying the brakes until the bicycle came to a complete stop.

Choice A is incorrect. The speed of Sheila’s bicycle before she applied the brakes is represented by the y-coordinate of the y-intercept of the graph of the given equation, not the x-coordinate of the x-intercept. Choice B is incorrect. The number of feet per second the speed of Sheila’s bicycle decreased each second after Sheila applied the brakes is represented by the slope of the graph of the given equation, not the x-coordinate of the x-intercept. Choice D is incorrect and may result from misinterpreting x as the distance, in feet, traveled after applying the brakes, rather than the time, in seconds, after applying the brakes.

 

Question 434 434 of 569 selected Linear Functions E

To repair a refrigerator, a technician charges $60 per hour for labor plus $120 for parts. Which function f represents the total amount, in dollars, the technician will charge for this job if it takes x hours?

  1. f(x)=x+120

  2. f(x)=60x

  3. f(x)=60x+120

  4. f(x)=60x-120

Show Answer Correct Answer: C

Choice C is correct. It’s given that the technician charges $60 per hour for labor. Therefore, if the job takes x hours, the technician will charge ($601 hour)(x hours), or $60x, for labor. It’s also given that the technician charges $120 for parts. Therefore, f(x)=60x+120 represents the total amount, in dollars, the technician will charge for this job if it takes x hours.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect. This function represents the total amount, in dollars, the technician charges for labor only, not the total amount charged for labor and parts.

Choice D is incorrect. This function represents the total amount, in dollars, the technician would charge if the charge for parts were subtracted from, rather than added to, the charge for labor.

Question 435 435 of 569 selected Linear Equations In 2 Variables E

A teacher is creating an assignment worth 70 points. The assignment will consist of questions worth 1 point and questions worth 3 points. Which equation represents this situation, where x represents the number of 1 -point questions and y represents the number of 3 -point questions?

  1. 4 x y = 70

  2. 4(x+y)=70

  3. 3 x + y = 70

  4. x + 3 y = 70

Show Answer Correct Answer: D

Choice D is correct. Since x represents the number of 1 -point questions and y represents the number of 3 -point questions, the assignment is worth a total of 1·x+3·y, or x+3y, points. Since the assignment is worth 70 points, the equation x+3y=70 represents this situation.

Choice A is incorrect and may result from conceptual errors.

Choice B is incorrect and may result from conceptual errors.

Choice C is incorrect and may result from conceptual errors.

Question 436 436 of 569 selected Linear Equations In 2 Variables M

Figure A and figure B are both regular polygons. The sum of the perimeter of figure A and the perimeter of figure B is 63 inches. The equation 3 x + 6 y = 63 represents this situation, where x is the number of sides of figure A and y is the number of sides of figure B. Which statement is the best interpretation of 6 in this context?

  1. Each side of figure B has a length of 6 inches.

  2. The number of sides of figure B is 6 .

  3. Each side of figure A has a length of 6 inches.

  4. The number of sides of figure A is 6 .

Show Answer Correct Answer: A

Choice A is correct. It’s given that figure A and figure B (not shown) are both regular polygons and the sum of the perimeters of the two figures is 63 inches. It’s also given that x is the number of sides of figure A and y is the number of sides of figure B, and that the equation 3x+6y=63 represents this situation. Thus, 3x and 6y represent the perimeters, in inches, of figure A and figure B, respectively. Since 6y represents the perimeter, in inches, of figure B and y is the number of sides of figure B, it follows that each side of figure B has a length of 6 inches.

Choice B is incorrect. The number of sides of figure B is y , not 6 .

Choice C is incorrect. Since the perimeter, in inches, of figure A is represented by 3x, each side of figure A has a length of 3 inches, not 6 inches.

Choice D is incorrect. The number of sides of figure A is x , not 6 .

Question 437 437 of 569 selected Systems Of 2 Linear Equations In 2 Variables M

y equals, 2 x minus 3, and, 3 y equals 5 x

In the solution to the system of equations above, what is the value of y ?

  1. negative 15

  2. negative 9

  3. 9

  4. 15

Show Answer Correct Answer: D

Choice D is correct. Multiplying both sides of y equals 2 x minus 3 by 5 results in 5 y, equals 10 x minus 15. Multiplying both sides of 3 y, equals 5 x by 2 results in 6 y, equals 10 x. Subtracting the resulting equations yields 5 y minus 6 y, equals, open parenthesis, 10 x minus 15, close parenthesis, minus, 10 x, which simplifies to negative y equals negative 15. Dividing both sides of negative y equals negative 15 by negative 1 results in y equals 15.

Choices A and B are incorrect and may result from incorrectly subtracting the transformed equation. Choice C is incorrect and may result from finding the value of x instead of the value of y.

 

Question 438 438 of 569 selected Linear Inequalities In 1 Or 2 Variables E

An elementary school teacher is ordering x workbooks and y sets of flash cards for a math class. The teacher must order at least 20 items, but the total cost of the order must not be over $80. If the workbooks cost $3 each and the flash cards cost $4 per set, which of the following systems of inequalities models this situation?

  1. x plus y, is greater than or equal to 20, and, 3 x plus 4 y, is less than or equal to 80

  2. x plus y, is greater than or equal to 20, and, 3 x plus 4 y, is greater than or equal to 80

  3. 3 x plus 4 y, is less than or equal to 20, and, x plus y, is greater than or equal to 80

  4. x plus y, is less than or equal to 20, and, 3 x plus 4 y, is greater than or equal to 80

Show Answer Correct Answer: A

Choice A is correct. The total number of workbooks and sets of flash cards ordered is represented by x + y. Since the teacher must order at least 20 items, it must be true that x + y ≥ 20. Each workbook costs $3; therefore, 3x represents the cost, in dollars, of x workbooks. Each set of flashcards costs $4; therefore, 4y represents the cost, in dollars, of y sets of flashcards. It follows that the total cost for x workbooks and y sets of flashcards is 3x + 4y. Since the total cost of the order must not be over $80, it must also be true that 3x + 4y ≤ 80. Of the choices given, these inequalities are shown only in choice A.


Choice B is incorrect. The second inequality says that the total cost must be greater, not less than or equal to $80. Choice C incorrectly limits the cost by the minimum number of items and the number of items with the maximum cost. Choice D is incorrect. The first inequality incorrectly says that at most 20 items must be ordered, and the second inequality says that the total cost of the order must be at least, not at most, $80.

 

Question 439 439 of 569 selected Systems Of 2 Linear Equations In 2 Variables H

78y-58x=47-78y

54x+74=py+154


In the given system of equations, p is a constant. If the system has no solution, what is the value of p ?

Show Answer Correct Answer: 3.5, 7/2

The correct answer is 72. A system of two linear equations in two variables, x and y , has no solution if the lines represented by the equations in the xy-plane are distinct and parallel. Two lines represented by equations in standard form Ax+By=C, where A , B , and C are constants, are parallel if the coefficients for x and y in one equation are proportional to the corresponding coefficients in the other equation. The first equation in the given system, 78y-58x=47-78y, can be written in standard form by adding 78y to both sides of the equation, which yields 148y-58x=47, or -58x+148y=47. Multiplying each term in this equation by -8 yields 5x-14y=-327. The second equation in the given system, 54x+74=py+154, can be written in standard form by subtracting 74 and py from both sides of the equation, which yields 54x-py=84. Multiplying each term in this equation by 4 yields 5x-4py=8. The coefficient of x in the first equation, 5x-14y=-327, is equal to the coefficient of x in the second equation, 5x-4py=8. For the lines to be parallel, and for the coefficients for x and y in one equation to be proportional to the corresponding coefficients in the other equation, the coefficient of y in the second equation must also be equal to the coefficient of y in the first equation. Therefore, -14=-4p. Dividing both sides of this equation by -4 yields -14-4=p, or p=72. Therefore, if the given system of equations has no solution, the value of p is 72. Note that 7/2 and 3.5 are examples of ways to enter a correct answer.

Question 440 440 of 569 selected Linear Equations In 2 Variables E

A shipment consists of 5 -pound boxes and 10 -pound boxes with a total weight of 220 pounds. There are 13 10 -pound boxes in the shipment. How many 5 -pound boxes are in the shipment?

  1. 5

  2. 10

  3. 13

  4. 18

Show Answer Correct Answer: D

Choice D is correct. It's given that the shipment consists of 5 -pound boxes and 10 -pound boxes with a total weight of 220 pounds. Let x represent the number of 5 -pound boxes and y represent the number of 10 -pound boxes in the shipment. Therefore, the equation 5 x + 10 y = 220 represents this situation. It's given that there are 13 10 -pound boxes in the shipment. Substituting 13 for y in the equation 5 x + 10 y = 220 yields 5x+10(13)=220, or 5 x + 130 = 220 . Subtracting 130 from both sides of this equation yields 5 x = 90 . Dividing both sides of this equation by 5 yields 18 . Thus, there are 18 5 -pound boxes in the shipment.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect. This is the number of 10 -pound boxes in the shipment.

Question 441 441 of 569 selected Systems Of 2 Linear Equations In 2 Variables H

4 x - 9 y = 9 y + 5

hy=2+4x

In the given system of equations, h is a constant. If the system has no solution, what is the value of h ?

  1. -9

  2. 0

  3. 9

  4. 18

Show Answer Correct Answer: D

Choice D is correct. A system of two linear equations in two variables, x and y , has no solution if the lines represented by the equations in the xy-plane are distinct and parallel. The graphs of two lines in the xy-plane represented by equations in the form Ax+By=C, where A , B , and C are constants, are parallel if the coefficients for x and y in one equation are proportional to the corresponding coefficients in the other equation. The first equation in the given system can be written in the form Ax+By=C by subtracting 9 y from both sides of the equation to yield 4x-18y=5. The second equation in the given system can be written in the form Ax+By=C by subtracting 4 x from both sides of the equation to yield -4x+hy=2. The coefficient of x in this second equation, -4, is -1 times the coefficient of x in the first equation, 4 . For the lines to be parallel, the coefficient of y in the second equation, h , must also be -1 times the coefficient of y in the first equation, -18. Thus, h=-1(-18), or h=18. Therefore, if the given system has no solution, the value of h is 18 .

Choice A is incorrect. If the value of h is -9 , then the given system would have one solution, rather than no solution.

Choice B is incorrect. If the value of h is 0 , then the given system would have one solution, rather than no solution.

Choice C is incorrect. If the value of h is 9 , then the given system would have one solution, rather than no solution.

Question 442 442 of 569 selected Linear Inequalities In 1 Or 2 Variables E

A geologist estimates that the volume of a slab of granite is greater than 12.7 cubic feet but less than 15.7 cubic feet. The geologist also estimates that the slab of granite weighs 165 pounds per cubic foot of volume. Which inequality represents this situation, where x represents the estimated total weight, in pounds, of the slab of granite?

  1. 165-15.7<x<165-12.7

  2. 165+12.7<x<165+15.7

  3. 165(12.7)<x<165(15.7)

  4. 16515.7<x<16512.7

Show Answer Correct Answer: C

Choice C is correct. It's given that the estimated volume of the slab of granite is greater than 12.7 cubic feet but less than 15.7 cubic feet. It's also given that the estimated weight per cubic foot of volume of that slab is 165 pounds. The estimated total weight of the slab of granite, in pounds, can be calculated by multiplying the estimated volume by the estimated weight per cubic foot. Therefore, if the estimated volume of the slab of granite is 12.7 cubic feet, its estimated total weight is 165(12.7) pounds, and if the estimated volume of the slab of granite is 15.7 cubic feet, its estimated total weight is 165(15.7) pounds. Since the estimated volume of the slab of granite is greater than 12.7 cubic feet but less than 15.7 cubic feet, the estimated total weight x, in pounds, must be greater than 165(12.7) pounds and less than 165(15.7) pounds. This situation can be represented by the inequality 165(12.7)<x<165(15.7).

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 443 443 of 569 selected Linear Functions E

In the linear function h h(0)=41 and h(1)=40. Which equation defines h ?

  1. h(x)=-x+41

  2. h(x)=-x

  3. h(x)=-41x

  4. h(x)=-41

Show Answer Correct Answer: A

Choice A is correct. An equation defining a linear function can be written in the form h(x)=ax+b, where a and b are constants. It’s given that h(0)=41. Substituting 0 for x and 41 for h(x) in the equation h(x)=ax+b yields 41=a(0)+b, or b=41. Substituting 41 for b in the equation h(x)=ax+b yields h(x)=ax+41. It’s also given that h(1)=40. Substituting 1 for x and 40 for h(x) in the equation h(x)=ax+41 yields 40=a(1)+41, or 40=a+41. Subtracting 41 from the left- and right-hand sides of this equation yields -1=a. Substituting -1 for a in the equation h(x)=ax+41 yields h(x)=-1x+41, or h(x)=-x+41.

Choice B is incorrect. Substituting 0 for x and 41 for h(x) in this equation yields 41=-0, which isn't a true statement.

Choice C is incorrect. Substituting 0 for x and 41 for h(x) in this equation yields 41=-41(0), or 41=0, which isn't a true statement.

Choice D is incorrect. Substituting 41 for h(x) in this equation yields 41=-41, which isn't a true statement.

Question 444 444 of 569 selected Linear Equations In 2 Variables M

2.5b+5r=80

The given equation describes the relationship between the number of birds, b , and the number of reptiles, r , that can be cared for at a pet care business on a given day. If the business cares for 16 reptiles on a given day, how many birds can it care for on this day?

  1. 0

  2. 5

  3. 40

  4. 80

Show Answer Correct Answer: A

Choice A is correct. The number of birds can be found by calculating the value of b when r = 16 in the given equation. Substituting 16 for r in the given equation yields 2.5b+5(16)=80, or 2.5 b + 80 = 80 . Subtracting 80 from both sides of this equation yields 2.5 b = 0 . Dividing both sides of this equation by 2.5 yields b = 0 . Therefore, if the business cares for 16 reptiles on a given day, it can care for 0 birds on this day.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 445 445 of 569 selected Linear Functions E

Hana deposited a fixed amount into her bank account each month. The function f(t)=100+25t gives the amount, in dollars, in Hana's bank account after t monthly deposits. What is the best interpretation of 25 in this context?

  1. With each monthly deposit, the amount in Hana's bank account increased by $25.

  2. Before Hana made any monthly deposits, the amount in her bank account was $25.

  3. After 1 monthly deposit, the amount in Hana's bank account was $25.

  4. Hana made a total of 25 monthly deposits.

Show Answer Correct Answer: A

Choice A is correct. It's given that t represents the number of monthly deposits. In the given function f(t)=100+25t, the coefficient of t is 25 . This means that for every increase in the value of t by 1 , the value of f(t) increases by 25 . It follows that with each monthly deposit, the amount in Hana's bank account increased by $25.

Choice B is incorrect. Before Hana made any monthly deposits, the amount in her bank account was $100

Choice C is incorrect. After 1 monthly deposit, the amount in Hana's bank account was $125

Choice D is incorrect and may result from conceptual errors.

Question 446 446 of 569 selected Systems Of 2 Linear Equations In 2 Variables E

  • For the first line in the system:
    • The line slants gradually down from left to right.
    • The line passes through the following points:
      • (0 comma 4)
      • (2 comma 2)
      • (4 comma 0)
  • For the second line in the system:
    • The line slants gradually up from left to right.
    • The line passes through the following points:
      • (0 comma 0)
      • (2 comma 2)
      • (4 comma 4)

The graph of a system of two linear equations is shown. What is the solution (x,y) to the system?

  1. (0,4)

  2. (2,2)

  3. (4,0)

  4. (4,4)

Show Answer Correct Answer: B

Choice B is correct. The solution to this system of linear equations is represented by the point that lies on both lines shown, or the point of intersection of the two lines. According to the graph, the point of intersection occurs when x = 2 and y = 2 , or at the point (2,2). Therefore, the solution (x,y) to the system is (2,2).

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 447 447 of 569 selected Linear Equations In 1 Variable H

- 49 x = - 98 x

How many solutions does the given equation have?

  1. Zero

  2. Exactly one

  3. Exactly two

  4. Infinitely many

Show Answer Correct Answer: B

Choice B is correct. Adding 98 x to each side of the given equation yields 49 x = 0 . Dividing each side of this equation by 49 yields x = 0 . This means that 0 is the only solution to the given equation. Therefore, the given equation has exactly one solution.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 448 448 of 569 selected Linear Inequalities In 1 Or 2 Variables M

An event planner is planning a party. It costs the event planner a onetime fee of $35 to rent the venue and $10.25 per attendee. The event planner has a budget of $200 . What is the greatest number of attendees possible without exceeding the budget?

Show Answer Correct Answer: 16

The correct answer is 16 . The total cost of the party is found by adding the onetime fee of the venue to the cost per attendee times the number of attendees. Let x be the number of attendees. The expression 35+10.25x thus represents the total cost of the party. It's given that the budget is $200, so this situation can be represented by the inequality 35+10.25x200. The greatest number of attendees can be found by solving this inequality for x . Subtracting 35 from both sides of this inequality gives 10.25x165. Dividing both sides of this inequality by 10.25 results in approximately x16.098. Since the question is stated in terms of attendees, rounding x down to the nearest whole number, 16 , gives the greatest number of attendees possible.

Question 449 449 of 569 selected Linear Equations In 1 Variable M

Megan’s regular wage at her job is p dollars per hour for the first 8 hours of work in a day plus 1.5 times her regular hourly wage for work in excess of 8 hours that day. On a given day, Megan worked for 10 hours, and her total earnings for that day were $137.50. What is Megan’s regular hourly wage?

  1. $11.75

  2. $12.50

  3. $13.25

  4. $13.75

Show Answer Correct Answer: B

Choice B is correct. Since p represents Megan’s regular pay per hour, 1.5p represents the pay per hour in excess of 8 hours. Since Megan worked for 10 hours, she must have been paid p dollars per hour for 8 of the hours plus 1.5p dollars per hour for the remaining 2 hours. Therefore, since Megan earned $137.50 for the 10 hours, the situation can be represented by the equation 137.5 = 8p + 2(1.5)p. Distributing the 2 in the equation gives 137.5 = 8p + 3p, and combining like terms gives 137.5 = 11p. Dividing both sides by 11 gives p = 12.5. Therefore, Megan’s regular wage is $12.50.

Choices A and C are incorrect and may be the result of calculation errors. Choice D is incorrect and may result from finding the average hourly wage that Megan earned for the 10 hours of work.

Question 450 450 of 569 selected Linear Equations In 2 Variables E

y = 70 x + 8

Which table gives three values of x and their corresponding values of y for the given equation?

Show Answer Correct Answer: A

Choice A is correct. Each of the given choices gives three values of x : 0 , 2 , and 4 . Substituting 0 for x in the given equation yields y=70(0)+8, or y = 8 . Therefore, when x = 0 , the corresponding value of y for the given equation is 8 . Substituting 2 for x in the given equation yields y=70(2)+8, or y = 148 . Therefore, when x = 2 , the corresponding value of y for the given equation is 148 . Substituting 4 for x in the given equation yields y=70(4)+8, or y = 288 . Therefore, when x = 4 , the corresponding value of y for the given equation is 288 . Thus, if the three values of x are 0 , 2 , and 4 , then their corresponding values of y are 8 , 148 , and 288 , respectively, for the given equation.

Choice B is incorrect. This table gives three values of x and their corresponding values of y for the equation y = 4 x + 70 .

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 451 451 of 569 selected Linear Functions E

The function f is defined by f(x)=5x+8. For what value of x does f(x)=58?

  1. 10

  2. 13

  3. 50

  4. 298

Show Answer Correct Answer: A

Choice A is correct. It's given that the function f is defined by f(x)=5x+8. Substituting 58 for f(x) in this equation yields 58=5x+8. Subtracting 8 from both sides of this equation yields 50 = 5 x . Dividing both sides of this equation by 5 yields 10 = x . Therefore, the value of x when f(x)=58 is 10 .

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect. This is the value of f(58), not the value of x when f(x)=58.

Question 452 452 of 569 selected Linear Equations In 1 Variable E

If 3 x plus 2 equals 8 , what is the value of 9 x plus 6 ?

Show Answer

The correct answer is 24. Multiplying both sides of the given equation by 3 yields 3 times, open parenthesis, 3 x plus 2, close parenthesis, equals 24. Using the distributive property to rewrite the left-hand side of this equation yields 9 x plus 6, equals 24.

Question 453 453 of 569 selected Linear Functions M

The graph of y=f(x)-11 is shown.

  • The line slants sharply down from left to right.
  • The line passes through the following points:
    • (negative 1 comma negative 2)
    • (0 comma negative 4)

Which equation defines the linear function f ?

  1. f(x)= - 13 x - 11

  2. f(x)= - 2 x + 7

  3. f(x)= - 13 x + 7

  4. f(x)= - 2 x - 11

Show Answer Correct Answer: B

Choice B is correct. The graph of a line in the xy-plane can be represented by the equation y=mx+b, where m is the slope of the line and (0,b) is the y-intercept. The slope of a line that passes through the points (x1,y1) and (x2,y2) can be calculated using the formula m=y2-y1x2-x1. The line shown passes through the points (-1,-2) and (0,-4). Substituting (-1,-2) and (0,4) for (x1,y1) and (x2,y2), respectively, in the formula m=y2-y1x2-x1 yields m=-4-(-2)0-(-1), which is equivalent to m=-21, or m=-2. Since the line shown passes through the point (0,-4), it follows that b=-4. Substituting - 2 for m and - 4 for b in the equation y=mx+b yields y=-2x-4. It’s given that the graph shown is the graph of y=f(x)-11. Substituting -2x-4 for y in the equation y=f(x)-11 yields -2x-4=f(x)-11. Adding 11 to both sides of this equation yields -2x+7=f(x). Therefore, the equation f(x)=-2x+7 defines the linear function f .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 454 454 of 569 selected Systems Of 2 Linear Equations In 2 Variables H

A piece of wire with a length of 32 inches is cut into two parts. One part has a length of x inches, and the other part has a length of y inches. The value of x is 4 more than 3 times the value of y . What is the value of x ?

Show Answer Correct Answer: 25

The correct answer is 25 . It’s given that a piece of wire has a length of 32 inches and is cut into two parts. It’s also given that one part has a length of x inches and the other part has a length of y inches. It follows that the equation x + y = 32 represents this situation. It’s also given that the value of x is 4 more than 3 times the value of y , or x=3y+4. Substituting 3y+4 for x in the equation x+y=32 yields 3y+4+y=32. Combining like terms on the left-hand side of this equation yields 4y+4=32. Subtracting 4 from both sides of this equation yields 4y=28. Dividing both sides of this equation by 4 yields y=7. Substituting 7 for y in the equation x=3y+4 yields x=3(7)+4, or x=25. Therefore, the value of x is 25 .

Question 455 455 of 569 selected Linear Equations In 1 Variable M

A bowl contains 20 ounces of water. When the bowl is uncovered, the amount of water in the bowl decreases by 1 ounce every 4 days. If 9 ounces of water remain in this bowl, for how many days has it been uncovered? 

  1. 3

  2. 7

  3. 36

  4. 44

Show Answer Correct Answer: D

Choice D is correct. It’s given that the bowl starts with 20 ounces of water and has 9 ounces of water remaining after a period of time has passed. The amount of water the bowl has lost during the time period can be found by subtracting the remaining amount of water from the amount of water the bowl starts with, which yields 20-9 ounces, or 11 ounces. This means the bowl loses 11 ounces of water during that period of time. It’s given that the amount of water decreases by 1 ounce every 4 days. Letting t represent the number of days the bowl has been uncovered, it follows that 14=11t. Multiplying both sides of this equation by 4 t yields t = 44 . Therefore, the bowl has been uncovered for 44 days. 

Choice A is incorrect and may result from conceptual or calculation errors. 

Choice B is incorrect and may result from conceptual or calculation errors. 

Choice C is incorrect. This is the value of t for the equation 14=9t, not 14=11t

Question 456 456 of 569 selected Linear Inequalities In 1 Or 2 Variables M

A model estimates that whales from the genus Eschrichtius travel 72 to 77 miles in the ocean each day during their migration. Based on this model, which inequality represents the estimated total number of miles, x , a whale from the genus Eschrichtius could travel in 16 days of its migration?

  1. 72+16x77+16

  2. (72)(16)x(77)(16)

  3. 7216+x77

  4. 7216x77

Show Answer Correct Answer: B

Choice B is correct. It's given that the model estimates that whales from the genus Eschrichtius travel 72 to 77 miles in the ocean each day during their migration. If one of these whales travels 72 miles each day for 16 days, then the whale travels 72(16) miles total. If one of these whales travels 77 miles each day for 16 days, then the whale travels 77(16) miles total. Therefore, the model estimates that in 16 days of its migration, a whale from the genus Eschrichtius could travel at least 72(16) and at most 77(16) miles total. Thus, the inequality (72)(16)x(77)(16) represents the estimated total number of miles, x , a whale from the genus Eschrichtius could travel in 16 days of its migration.

Choice A is incorrect and may result from conceptual errors.

Choice C is incorrect and may result from conceptual errors.

Choice D is incorrect and may result from conceptual errors.

Question 457 457 of 569 selected Linear Equations In 1 Variable E

k + 12 = 336

What is the solution to the given equation?

  1. 28

  2. 324

  3. 348

  4. 4,032

Show Answer Correct Answer: B

Choice B is correct. Subtracting 12 from both sides of the given equation yields k=324. Therefore, the solution to the given equation is 324 .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 458 458 of 569 selected Linear Functions H

The function f(x) is defined as 19 more than 4 times a number x . If y=f(x) is graphed in the x y -plane, what is the best interpretation of the x -intercept?

  1. When f(x)=0, the number is - 19 4 .

  2. When the number is 0 f(x)=19 .

  3. The value of f(x) increases by 1  for each increase of 4 in the value of the number.

  4. For each increase of 1 in the value of the number, f(x) increases by 4 .

Show Answer Correct Answer: A

Choice A is correct. It’s given that the function f(x) is defined as 19 more than 4 times a number x . This can be represented by the equation f(x)=4x+19. The x-intercept of the graph of y=f(x) in the xy-plane is the point where the graph intersects the x-axis, or the point on the graph where the value of f(x) is equal to 0 . Substituting 0 for f(x) in the equation f(x)=4x+19 yields 0=4x+19. Subtracting 19 from each side of this equation yields -19=4x. Dividing each side of this equation by 4 yields x=-194. Therefore, when f(x)=0, the number is -194

Choice B is incorrect. This is the best interpretation of the y-intercept, not the x-intercept, of the graph of the function.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect. This is the best interpretation of the slope, not the x-intercept, of the graph of the function.

Question 459 459 of 569 selected Linear Equations In 2 Variables H

The line with the equation four fifths x, plus one third y, equals 1 is graphed in the xy‑plane. What is the x-coordinate of the x‑intercept of the line?

Show Answer

The correct answer is 1.25. The y-coordinate of the x-intercept is 0, so 0 can be substituted for y, giving four fifths x, plus one third times 0, equals 1. This simplifies to four fifths x, equals 1. Multiplying both sides of four fifths x, equals 1 by 5 gives 4 x equals 5. Dividing both sides of 4 x equals 5 by 4 gives x equals five fourths, which is equivalent to 1.25. Note that 1.25 and 5/4 are examples of ways to enter a correct answer.

Question 460 460 of 569 selected Linear Functions E

The function h is defined by h(x)=x+200. What is the value of h(50)?

  1. 200

  2. 250

  3. 10,000

  4. 50,200

Show Answer Correct Answer: B

Choice B is correct. Substituting 50 for x in the given function yields h(50)=50+200, or h(50)=250. Therefore, the value of h(50) is 250 .

Choice A is incorrect. This is the value of h(0).

Choice C is incorrect. This is the value of h(9,800).

Choice D is incorrect. This is the value of h(50,000).

Question 461 461 of 569 selected Linear Equations In 2 Variables E

What is the y-intercept of the graph of y = 34 x + 81 in the xy-plane?

  1. (0,81)

  2. (0,34)

  3. (0,-34)

  4. (0,-81)

Show Answer Correct Answer: A

Choice A is correct. In the xy-plane, the graph of an equation in the form y = m x + b , where m and b are constants, has a slope of m and a y-intercept of (0,b). Therefore, the y-intercept of the graph of y = 34 x + 81 is (0,81).

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 462 462 of 569 selected Linear Functions E
Area (square feet) Water (gallons)
2,520 4,536
3,780 6,804
5,040 9,072

The buildings of a shopping center are designed to allow water to drain from the roof into gutters on the sides of the buildings. The table shows the relationship between the area x , in square feet, of a roof and the amount of water f(x), in gallons, drained from the roof into the gutters over a certain period of time. Which equation could define f ?

  1. f(x)= 0.6 x

  2. f(x)= 1.8 x

  3. f(x)= 2,268 x

  4. f(x)= 4,536 x

Show Answer Correct Answer: B

Choice B is correct. It's given that the table represents the relationship between the area x, in square feet, of the roof of a shopping center and the amount of water f(x), in gallons, drained from the roof into the gutters. Every choice represents this relationship with an equation defining f in the form f(x) = mx, where m is a constant rate of change. The value of m can be determined by dividing both sides of the equation by x. Each of three pairs of x and f(x) in the table yield m=1.8, since 4,5362,520=1.86,8043,780=1.8, and 9,0725,040=1.8. Therefore, the equation f(x)=1.8x could define f.

Choice A is incorrect. For the roof with an area of 2,520 square feet, this equation would yield 0.6(2,520), or 1,512, gallons, not the 4,536 gallons shown in the table.

Choice C is incorrect. For the roof with an area of 2,520 square feet, this equation would yield 2,268(2,520), or 5,715,360, gallons, not the 4,536 gallons shown in the table.

Choice D is incorrect. For the roof with an area of 2,520 square feet, this equation would yield 4,536(2,520), or 11,430,720, gallons, not the 4,536 gallons shown in the table.

Question 463 463 of 569 selected Systems Of 2 Linear Equations In 2 Variables H

One of the two equations in a linear system is 2 x plus 6 y, equals 10. The system has no solution. Which of the following could be the other equation in the system?

  1. x plus 3 y, equals 5

  2. x plus 3 y, equals negative 20

  3. 6 x minus 2 y, equals 0

  4. 6 x plus 2 y, equals 10

Show Answer Correct Answer: B

Choice B is correct. A system of two linear equations written in standard form has no solution when the equations are distinct and the ratio of the x-coefficient to the y-coefficient for one equation is equivalent to the ratio of the x-coefficient to the y-coefficient for the other equation. This ratio for the given equation is 2 to 6, or 1 to 3. Only choice B is an equation that isn’t equivalent to the given equation and whose ratio of the x-coefficient to the y-coefficient is 1 to 3.

Choice A is incorrect. Multiplying each of the terms in this equation by 2 yields an equation that is equivalent to the given equation. This system would have infinitely many solutions. Choices C and D are incorrect. The ratio of the x-coefficient to the y-coefficient in 6 x minus 2 y, equals 0 (choice C) is negative 6 to 2, or negative 3 to 1. This ratio in 6 x plus 2 y, equals 10 (choice D) is 6 to 2, or 3 to 1. Since neither of these ratios is equivalent to that for the given equation, these systems would have exactly one solution.

 

Question 464 464 of 569 selected Linear Equations In 2 Variables H

  • The line slants gradually down from left to right.
  • The line passes through the following points:
    • (0 comma 7)
    • (8 comma 0)

The point with coordinates (d , 4 ) lies on the line shown. What is the value of d ?

  1. 7 2

  2. 26 7

  3. 24 7

  4. 27 8

Show Answer Correct Answer: C

Choice C is correct. It's given from the graph that the points (0,7) and (8,0) lie on the line. For two points on a line, (x1,y1) and (x2,y2), the slope of the line can be calculated using the slope formula m=y2-y1x2-x1. Substituting (0,7) for (x1,y1) and (8,0) for (x2,y2) in this formula, the slope of the line can be calculated as m=0-78-0, or m=-78. It's also given that the point (d,4) lies on the line. Substituting (d,4) for (x1,y1), (8,0) for (x2,y2), and -78 for m in the slope formula yields -78=0-48-d, or -78=-48-d. Multiplying both sides of this equation by 8-d yields -78(8-d)=-4. Expanding the left-hand side of this equation yields -7+78d=-4. Adding 7 to both sides of this equation yields 78d=3. Multiplying both sides of this equation by 87 yields d=247. Thus, the value of d is 247.

Choice A is incorrect. This is the value of y when x=4.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 465 465 of 569 selected Linear Equations In 1 Variable H

If 5-7(2-4x)=16-8(2-4x), what is the value of 2-4x?

Show Answer Correct Answer: 11

The correct answer is 11 . Subtracting 5 from each side of the given equation yields -7(2-4x)=11-8(2-4x). Adding 8(2-4x) to each side of this equation yields 2-4x=11. Therefore, the value of 2-4x is 11 .

Question 466 466 of 569 selected Linear Equations In 1 Variable M

66x=66x

How many solutions does the given equation have?

  1. Exactly one

  2. Exactly two

  3. Infinitely many

  4. Zero

Show Answer Correct Answer: C

Choice C is correct. If the two sides of a linear equation are equivalent, then the equation is true for any value. If an equation is true for any value, it has infinitely many solutions. Since the two sides of the given linear equation 66x=66x are equivalent, the given equation has infinitely many solutions.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 467 467 of 569 selected Systems Of 2 Linear Equations In 2 Variables M

y=13x-14

y = - x + 18

The solution to the given system of equations is (x,y). What is the value of x ?

Show Answer Correct Answer: 24

The correct answer is 24 . The given system of equations can be solved by the substitution method. The first equation in the given system of equations is y=13x-14. Substituting 13x-14 for y in the second equation in the given system yields 13x-14=-x+18. Adding 14 to both sides of this equation yields 13x=-x+32. Adding x to both sides of this equation yields 43x=32. Multiplying both sides of this equation by 34 yields x = 24 .

Question 468 468 of 569 selected Linear Equations In 1 Variable M

4x+12=a(x+3)2

In the given equation, a is a constant. If the equation has infinitely many solutions, what is the value of a ?

  1. 0

  2. 3

  3. 8

  4. 12

Show Answer Correct Answer: C

Choice C is correct. If an equation has infinitely many solutions, then the two sides of the equation must be equivalent. Multiplying each side of the given equation by 2 yields  8 x + 24 = a ( x + 3 ) . Since 8 is a common factor of both terms on the left-hand side of this equation, the equation can be rewritten as 8(x+3)=a(x+3). The two sides of this equation are equivalent when a = 8 . Therefore, if the given equation has infinitely many solutions, the value of a is 8 .

Alternate approach: If the given equation, 4x+12=a(x+3)2, has infinitely many solutions, then both sides of this equation are equal for any value of x . If x = 0 , then substituting 0 for x in the given equation yields 4(0)+12=a(0+3)2, or 12=32a. Dividing both sides of this equation by 3 2 yields 8=a.

Choice A is incorrect. If the value of a is 0 , the given equation is equivalent to 4 x + 12 = 0 , which has one solution, not infinitely many solutions.

Choice B is incorrect. If the value of a is 3 , the given equation is equivalent to 4x+12=3(x+3)2, or 4x+12=32x+92, which has one solution, not infinitely many solutions.

Choice D is incorrect. If the value of a is 12 , the given equation is equivalent to 4x+12=12(x+3)2, or 4 x + 12 = 6 x + 18 , which has one solution, not infinitely many solutions.

Question 469 469 of 569 selected Linear Functions H

The figure presents a 3-column table, with 3 rows of data, titled “Energy per Gram of Typical Macronutrients.” The heading for column 1 is “Macronutrient,” the heading for column 2 is “Food calories,” and the heading for column 3 is “Kilojoules.” The 3 rows of data are as follows. Row 1.Macronutrient, Protein; Food calories, 4 point zero; Kilojoules, 16 point 7. Row 2. Macronutrient, Fat; Food calories, 9 point zero; Kilojoules, 37 point 7. Row 3. Macronutrient, Carbohydrate; Food calories, 4 point zero; Kilojoules, 16 point 7.

The table above gives the typical amounts of energy per gram, expressed in both food calories and kilojoules, of the three macronutrients in food. If the 180 food calories in a granola bar come entirely from p grams of protein, f grams of fat, and c grams of carbohydrate, which of the following expresses f in terms of p and c ?

 

  1. f equals, 20 plus the fraction 4 over 9, end fraction, times, open parenthesis, p plus c, close parenthesis

  2. f equals, 20 minus the fraction 4 over 9, end fraction, times, open parenthesis, p plus c, close parenthesis

  3. f equals, 20 minus the fraction 4 over 9, end fraction, times, open parenthesis, p minus c, close parenthesis

  4. f equals, 20 plus the fraction 9 over 4, end fraction, times, open parenthesis, p plus c, close parenthesis

Show Answer Correct Answer: B

Choice B is correct. It is given that there are 4.0 food calories per gram of protein, 9.0 food calories per gram of fat, and 4.0 food calories per gram of carbohydrate. If 180 food calories in a granola bar came from p grams of protein, f grams of fat, and c grams of carbohydrate, then the situation can be represented by the equation 180 equals, 4 p plus 9 f, plus 4 c. The equation can then be rewritten in terms of f by subtracting 4p and 4c from both sides of the equation and then dividing both sides of the equation by 9. The result is the equation f equals, 20 minus, four ninths times, open parenthesis, p plus c, close parenthesis.

Choices A, C, and D are incorrect and may be the result of not representing the situation with the correct equation or incorrectly rewriting the equation in terms of f.

 

Question 470 470 of 569 selected Systems Of 2 Linear Equations In 2 Variables M
The figure presents the graph of two intersecting lines in the x y plane with the origin labeled O. The integers negative 5 through 5 are indicated on each axis. One line begins above the x axis and to the left of the y axis, and trends downward and to the right. It crosses the y axis between 0 and 1, then crosses the x axis between 1 and 2, and ends below the x axis and to the right of the y axis. The other line begins below the x axis and slightly to the left of the y axis, and trends upward and to the right. It crosses the y axis at negative 5, then crosses the x axis between 1 and 2, and ends above the x axis and to the right of the y axis. The two lines intersect at the point where both lines cross the x axis between 1 and 2.

Which of the following systems of equations has the same solution as the system of equations graphed above?

  1. Equation 1: y equals 0. Equation 2: x equals three halves

  2. Equation 1:  y equals three halves. Equation 2: x equals 0

  3. Equation 1: y equals 0. Equation 2: x equals 1

  4. Equation 1: y equals 1. Equation 2: x equals 0

Show Answer Correct Answer: A

Choice A is correct. The solution to a system of equations is the coordinates of the intersection point of the graphs of the equations in the xy-plane. Based on the graph, the solution to the given system of equations is best approximated as the point with coordinates three halves comma 0. In the xy-plane, the graph of y equals 0 is a horizontal line on which every y-coordinate is 0, and the graph of x equals three halves is a vertical line on which every x-coordinate is three halves. These graphs intersect at the point with coordinates three halves comma 0. Therefore, the system of equations in choice A has the same solution as the given system.

Choices B, C, and D are incorrect. If graphed in the xy-plane, these choices would intersect at the points with coordinates 0 comma three halves, 1 comma 0, and 0 comma 1, respectively, not the point with coordinates three halves comma 0.

 

Question 471 471 of 569 selected Linear Inequalities In 1 Or 2 Variables M

A moving truck can tow a trailer if the combined weight of the trailer and the boxes it contains is no more than 4,600 pounds. What is the maximum number of boxes this truck can tow in a trailer with a weight of 500 pounds if each box weighs 120 pounds?

  1. 34

  2. 35

  3. 38

  4. 39

Show Answer Correct Answer: A

Choice A is correct. It’s given that the truck can tow a trailer if the combined weight of the trailer and the boxes it contains is no more than 4,600 pounds. If the trailer has a weight of 500 pounds and each box weighs 120 pounds, the expression 500+120b, where b is the number of boxes, gives the combined weight of the trailer and the boxes. Since the combined weight must be no more than 4,600 pounds, the possible numbers of boxes the truck can tow are given by the inequality 500+120b4,600. Subtracting 500 from both sides of this inequality yields 120b4,100. Dividing both sides of this inequality by 120 yields b2056, or b is less than or equal to approximately 34.17 . Since the number of boxes, b , must be a whole number, the maximum number of boxes the truck can tow is the greatest whole number less than 34.17 , which is 34 .

Choice B is incorrect. Towing the trailer and 35 boxes would yield a combined weight of 4,700 pounds, which is greater than 4,600 pounds.

Choice C is incorrect. Towing the trailer and 38 boxes would yield a combined weight of 5,060 pounds, which is greater than 4,600 pounds.

Choice D is incorrect. Towing the trailer and 39 boxes would yield a combined weight of 5,180 pounds, which is greater than 4,600 pounds.

Question 472 472 of 569 selected Linear Functions H

An object hangs from a spring. The formula l equals, 30 plus 2 w relates the length l, in centimeters, of the spring to the weight w, in newtons, of the object. Which of the following describes the meaning of the 2 in this context?

  1. The length, in centimeters, of the spring with no weight attached

  2. The weight, in newtons, of an object that will stretch the spring 30 centimeters

  3. The increase in the weight, in newtons, of the object for each one-centimeter increase in the length of the spring

  4. The increase in the length, in centimeters, of the spring for each one-newton increase in the weight of the object

Show Answer Correct Answer: D

Choice D is correct. The value 2 is multiplied by w, the weight of the object. When the weight is 0, the length is 30 + 2(0) = 30 centimeters. If the weight increases by w newtons, the length increases by 2w centimeters, or 2 centimeters for each one-newton increase in weight.


Choice A is incorrect because this describes the value 30. Choice B is incorrect because 30 represents the length of the spring before it has been stretched. Choice C is incorrect because this describes the value w.

Question 473 473 of 569 selected Linear Equations In 1 Variable H

The equation 9 x plus 5, equals a, times, open parenthesis, x plus b, close parenthesis, where a and b are constants, has no solutions. Which of the following must be true?

I. a, equals 9

II. b equals 5

III. b is not equal to five ninths

  1. None

  2. I only

  3. I and II only

  4. I and III only

Show Answer Correct Answer: D

Choice D is correct. For a linear equation in a form a, x plus b, equals, c x plus d to have no solutions, the x-terms must have equal coefficients and the remaining terms must not be equal. Expanding the right-hand side of the given equation yields 9 x plus 5, equals, a, x plus a, b. Inspecting the x-terms, 9 must equal a, so statement I must be true. Inspecting the remaining terms, 5 can’t equal 9 b. Dividing both of these quantities by 9 yields that b can’t equal five ninths. Therefore, statement III must be true. Since b can have any value other than five ninths, statement II may or may not be true.

Choice A is incorrect. For the given equation to have no solution, both a, equals 9 and b is not equal to five ninths must be true. Choice B is incorrect because it must also be true that b is not equal to five ninths . Choice C is incorrect because when a, equals 9 , there are many values of b that lead to an equation having no solution. That is, b might be 5, but b isn’t required to be 5.

Question 474 474 of 569 selected Linear Equations In 2 Variables E

  • The line slants gradually up from left to right.
  • The line passes through the following points:
    • (negative 9 comma 0)
    • (0 comma 5)

What is the y-intercept of the line graphed?

  1. (-5,0)

  2. (0,0)

  3. (0,5)

  4. (0,9)

Show Answer Correct Answer: C

Choice C is correct. The y-intercept of a graph is the point where the graph intersects the y-axis. The line graphed intersects the y-axis at the point (0,5). Therefore, the y-intercept of the line graphed is (0,5).

Choice A is incorrect and may result from conceptual errors.

Choice B is incorrect and may result from conceptual errors.

Choice D is incorrect and may result from conceptual errors.

Question 475 475 of 569 selected Systems Of 2 Linear Equations In 2 Variables E

Equation 1: 3 x plus y, equals 29. Equation 2: x equals 2

If the ordered pair x comma y is the solution to the given system of equations, what is the value of y ?

Show Answer

The correct answer is 23. Since it’s given that x equals 2, the value of y can be found by substituting 2 for x in the first equation and solving for y. Substituting 2 for x yields 3 times 2, plus y, equals 29, or 6 plus y, equals 29. Subtracting 6 from both sides of this equation yields y equals 23.

Question 476 476 of 569 selected Systems Of 2 Linear Equations In 2 Variables H

(x-2)-4(y+7)=117

(x-2)+4(y+7)=442

The solution to the given system of equations is (x,y). What is the value of 6(x-2)?

Show Answer Correct Answer: 1677

The correct answer is 1,677 . Adding the first equation to the second equation in the given system yields (x-2)+(x-2)+(-4)(y+7)+4(y+7)=117+442, or 2(x-2)=559. Multiplying both sides of this equation by 3 yields 6(x-2)=1,677. Therefore, the value of 6(x-2) is 1,677 .

Question 477 477 of 569 selected Linear Equations In 2 Variables E

F equals, 2 point 5 0 x, plus 7 point 0 0 y

In the equation above, F represents the total amount of money, in dollars, a food truck charges for x drinks and y salads. The price, in dollars, of each drink is the same, and the price, in dollars, of each salad is the same. Which of the following is the best interpretation for the number 7.00 in this context?

  1. The price, in dollars, of one drink

  2. The price, in dollars, of one salad

  3. The number of drinks bought during the day

  4. The number of salads bought during the day

Show Answer Correct Answer: B

Choice B is correct. It’s given that 2 point 5 0 x, plus 7 point 0 0 y is equal to the total amount of money, in dollars, a food truck charges for x drinks and y salads. Since each salad has the same price, it follows that the total charge for y salads is 7 point 0 0 y dollars. When y equals 1, the value of the expression 7 point 0 0 y is 7 point 0 0 times 1, or 7.00. Therefore, the price for one salad is 7.00 dollars.

Choice A is incorrect. Since each drink has the same price, it follows that the total charge for x drinks is 2 point 5 0 x dollars. Therefore, the price, in dollars, for one drink is 2.50, not 7.00. Choices C and D are incorrect. In the given equation, F represents the total charge, in dollars, when x drinks and y salads are bought at the food truck. No information is provided about the number of drinks or the number of salads that are bought during the day. Therefore, 7.00 doesn’t represent either of these quantities.

 

Question 478 478 of 569 selected Linear Functions H

For the function f, if f of open parenthesis, 3 x, close parenthesis, equals x minus 6 for all values of x, what is the value of f of 6 ?

  1. negative 6

  2. negative 4

  3. 0

  4. 2

Show Answer Correct Answer: B

Choice B is correct. It’s given that f of 3 x, equals x minus 6 for all values of x. If 3 x equals 6, then f of 3 x will equal f of 6. Dividing both sides of 3 x equals 6 by 3 gives x equals 2. Therefore, substituting 2 for x in the given equation yields f of, open parenthesis, 3 times 2, close parenthesis, equals 2 minus 6, which can be rewritten as f of 6, equals negative 4.

Choice A is incorrect. This is the value of the constant in the given equation for f. Choice C is incorrect and may result from substituting x equals 6, rather than x equals 2, into the given equation. Choice D is incorrect. This is the value of x that yields f of 6 for the left-hand side of the given equation; it’s not the value of f of 6 .

 

Question 479 479 of 569 selected Systems Of 2 Linear Equations In 2 Variables M

3 y = 4 x + 17

- 3 y = 9 x - 23

The solution to the given system of equations is (x,y). What is the value of 39 x ?

  1. -18

  2. -6

  3. 6

  4. 18

Show Answer Correct Answer: D

Choice D is correct. Adding the second equation to the first equation in the given system of equations yields 3y-3y=4x+9x+17-23, or 0 = 13 x - 6 . Adding 6 to each side of this equation yields 6 = 13 x . Multiplying each side of this equation by 3 yields 18 = 39 x . Therefore, the value of 39 x is 18 .

Choice A is incorrect. This is the value of - 39 x , not 39 x .

Choice B is incorrect. This is the value of - 13 x , not 39 x .

Choice C is incorrect. This is the value of 13 x , not 39 x .

Question 480 480 of 569 selected Systems Of 2 Linear Equations In 2 Variables M

A proposal for a new library was included on an election ballot. A radio show stated that 3 times as many people voted in favor of the proposal as people who voted against it. A social media post reported that 15,000 more people voted in favor of the proposal than voted against it. Based on these data, how many people voted against the proposal?

  1. 7,500

  2. 15,000

  3. 22,500

  4. 45,000

Show Answer Correct Answer: A

Choice A is correct. It's given that a radio show stated that 3 times as many people voted in favor of the proposal as people who voted against it. Let x represent the number of people who voted against the proposal. It follows that 3 x is the number of people who voted in favor of the proposal and 3x-x, or 2 x , is how many more people voted in favor of the proposal than voted against it. It's also given that a social media post reported that 15,000 more people voted in favor of the proposal than voted against it. Thus, 2x=15,000. Since 2x=15,000, the value of x must be half of 15,000 , or 7,500 . Therefore, 7,500 people voted against the proposal.

Choice B is incorrect. This is how many more people voted in favor of the proposal than voted against it, not the number of people who voted against the proposal.

Choice C is incorrect. This is the number of people who voted in favor of the proposal, not the number of people who voted against the proposal.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 481 481 of 569 selected Linear Equations In 1 Variable E

A rocket contained 467,000 kilograms (kg) of propellant before launch. Exactly 21 seconds after launch, 362,105 kg of this propellant remained. On average, approximately how much propellant, in kg, did the rocket burn each second after launch?

  1. 4,995

  2. 17,243

  3. 39,481

  4. 104,895

Show Answer Correct Answer: A

Choice A is correct. It’s given that the rocket contained 467,000 kilograms (kg) of propellant before launch and had 362,105 kg remaining exactly 21 seconds after launch. Finding the difference between the amount, in kg, of propellant before launch and the remaining amount, in kg, of propellant after launch gives the amount, in kg, of propellant burned during the 21 seconds: 467,000-362,105=104,895. Dividing the amount of propellant burned by the number of seconds yields 104,89521=4,995. Thus, an average of 4,995 kg of propellant burned each second after launch.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from finding the amount of propellant burned, rather than the amount of propellant burned each second.

Question 482 482 of 569 selected Linear Functions H

The linear function g is defined by g(x)=b-15x, where b is a constant. If g(c+7)=c4, where c is a constant, which of the following expressions represents the value of b ?

  1. 15 c 4

  2. 19 c 4 + 7

  3. 61 c 4 + 105

  4. 15 c + 105

Show Answer Correct Answer: C

Choice C is correct. It’s given that g(c+7)=c4. Therefore, for the given linear function g , when x = c + 7 , g(x)=c4. Substituting c + 7 for x and c4 for g(x) in g(x)=b-15x yields c4=b-15(c+7). Applying the distributive property to the right-hand side of this equation yields c4=b-15c-105. Adding 15 c to both sides of this equation yields c4+15c=b-105. Adding 105 to both sides of this equation yields c4+15c+105=b, or 61c4+105=b. Therefore, the expression that represents the value of b is 61c4+105.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 483 483 of 569 selected Systems Of 2 Linear Equations In 2 Variables H

2x+3y=7
10x+15y=35

For each real number r , which of the following points lies on the graph of each equation in the xy-plane for the given system?

  1. (r5+7,-r5+35)

  2. (-3r2+72, r)

  3. (r, 2r3+73)

  4. (r,-3r2+72)

Show Answer Correct Answer: B

Choice B is correct. The two given equations are equivalent because the second equation can be obtained from the first equation by multiplying each side of the equation by 5 . Thus, the graphs of the equations are coincident, so if a point lies on the graph of one of the equations, it also lies on the graph of the other equation. A point (x,y) lies on the graph of an equation in the xy-plane if and only if this point represents a solution to the equation. It is sufficient, therefore, to find the point that represents a solution to the first given equation. Substituting the x- and y-coordinates of choice B, -3r2+72 and r , for x and y , respectively, in the first equation yields 2(-3r2+72)+3r=7, which is equivalent to -3r+7+3r=7, or 7=7. Therefore, the point (-3r2+72,r) represents a solution to the first equation and thus lies on the graph of each equation in the xy-plane for the given system.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 484 484 of 569 selected Linear Functions M

The function f is defined by f(x)=97x+87. For what value of x does f(x)=5?

Show Answer Correct Answer: 3

The correct answer is 3 . Substituting 5 for f(x) in the given function yields 5=97x+87. Multiplying each side of this equation by 7 yields 7(5)=7(97x+87), or 35=9x+8. Subtracting 8 from each side of this equation yields 27=9x. Dividing each side of this equation by 9 yields 3=x. Therefore, f(x)=5 when the value of x is 3 .

Question 485 485 of 569 selected Linear Equations In 2 Variables H

In the xy-plane, line k is defined by x plus y equals 0. Line j is perpendicular to line k, and the y-intercept of line j is the point with coordinates 0 comma 3. Which of the following is an equation of line j ?

  1. x plus y, equals 3

  2. x plus y, equals negative 3

  3. x minus y, equals 3

  4. x minus y, equals negative 3

Show Answer Correct Answer: D

Choice D is correct. It’s given that line j is perpendicular to line k and that line k is defined by the equation x plus y, equals 0. This equation can be rewritten in slope-intercept form, y equals, m x plus b, where m represents the slope of the line and b represents the y-coordinate of the y-intercept of the line, by subtracting x from both sides of the equation, which yields y equals, negative x. Thus, the slope of line k is negative 1. Since line j and line k are perpendicular, their slopes are opposite reciprocals of each other. Thus, the slope of line j is 1. It’s given that the y-intercept of line j is the point with coordinates 0 comma 3. Therefore, the equation for line j in slope-intercept form is y equals, x plus 3, which can be rewritten as x minus y, equals negative 3.

Choices A, B, and C are incorrect and may result from conceptual or calculation errors.

 

Question 486 486 of 569 selected Systems Of 2 Linear Equations In 2 Variables E

  • For the first line in the system:
    • The line slants sharply down from left to right.
    • The line passes through the following points:
      • (0 comma 3)
      • (4 comma negative 5)
  • For the second line in the system:
    • The line slants gradually down from left to right.
    • The line passes through the following points:
      • (0 comma negative 2)
      • (4 comma negative 5)

The graph of a system of linear equations is shown. What is the solution (x,y) to the system?

  1. (4,-5)

  2. (0,3)

  3. (0,-2)

  4. (-2,3)

Show Answer Correct Answer: A

Choice A is correct. The solution to this system of linear equations is represented by the point that lies on both lines shown, or the point of intersection of the two lines. According to the graph, the point of intersection occurs when x = 4 and y = - 5 , or at the point (4,-5). Therefore, the solution (x,y) to the system is (4,-5).

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 487 487 of 569 selected Linear Equations In 2 Variables E

A line in the xy-plane has a slope of -12 and passes through the point (0,3). Which equation represents this line?

  1. y=-12x-3

  2. y=-12x+3

  3. y=12x-3

  4. y=12x+3

Show Answer Correct Answer: B

Choice B is correct. A line in the xy-plane with a slope of m and a y-intercept of (0,b) can be represented by the equation y=mx+b. It's given that the line has a slope of -12. Therefore, m=-12. It's also given that the line passes through the point (0,3). Therefore, b=3. Substituting -12 for m and 3 for b in the equation y=mx+b yields y=-12x+3. Therefore, the equation y=-12x+3 represents this line.

Choice A is incorrect. This equation represents a line in the xy-plane that passes through the point (0,-3), not (0,3).

Choice C is incorrect. This equation represents a line in the xy-plane that has a slope of 12, not -12, and passes through the point (0,-3), not (0,3).

Choice D is incorrect. This equation represents a line in the xy-plane that has a slope of 12, not -12.

Question 488 488 of 569 selected Linear Inequalities In 1 Or 2 Variables M

The average annual energy cost for a certain home is $4,334. The homeowner plans to spend $25,000 to install a geothermal heating system. The homeowner estimates that the average annual energy cost will then be $2,712. Which of the following inequalities can be solved to find t, the number of years after installation at which the total amount of energy cost savings will exceed the installation cost?

  1. 25,000 is greater than, open parenthesis, 4,334 minus 2,712, close parenthesis, times t

  2. 25,000 is less than, open parenthesis, 4,334 minus 2,712, close parenthesis, times t

  3. 25,000 minus 4,334, is greater than 2,712 t

  4. 25,000 is greater than the fraction with numerator 4,332, and denominator 2,712, end fraction, t

Show Answer Correct Answer: B

Choice B is correct. The savings each year from installing the geothermal heating system will be the average annual energy cost for the home before the geothermal heating system installation minus the average annual energy cost after the geothermal heating system installation, which is 4,334 minus 2,712 dollars. In t years, the savings will be open parenthesis, 4,334 minus 2,712, close parenthesis, times t dollars. Therefore, the inequality that can be solved to find the number of years after installation at which the total amount of energy cost savings will exceed (be greater than) the installation cost, $25,000, is 25,000 is less than, open parenthesis, 4,334 minus 2,712, close parenthesis, times t.

Choice A is incorrect. It gives the number of years after installation at which the total amount of energy cost savings will be less than the installation cost. Choice C is incorrect and may result from subtracting the average annual energy cost for the home from the onetime cost
of the geothermal heating system installation. To find the predicted total savings, the predicted average cost should be subtracted from the average annual energy cost before the installation, and the result should be multiplied by the number of years, t. Choice D is incorrect and may result from misunderstanding the context. The ratio 4,332 over 2,712 compares the average energy cost before installation and the average energy cost after installation; it does not represent the savings.

 

Question 489 489 of 569 selected Linear Equations In 1 Variable E

Henry receives a $60.00 gift card to pay for movies online. He uses his gift card to buy 3 movies for $7.50 each. If he spends the rest of his gift card balance on renting movies for $1.50 each, how many movies can Henry rent?

  1. 10

  2. 25

  3. 35

  4. 40

Show Answer Correct Answer: B

Choice B is correct. It's given that Henry uses his $60.00 gift card to buy 3 movies for $7.50 each. Therefore, Henry spends 3($7.50), or $22.50, of his $60.00 gift card to buy 3 movies. After buying 3 movies with his $60.00 gift card, Henry has a gift card balance of $60.00-$22.50, or $37.50. It's also given that Henry spends the rest of his gift card balance on renting movies for $1.50 each. Therefore, Henry can rent $37.50$1.50, or 25 , movies.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 490 490 of 569 selected Linear Equations In 1 Variable H

What value of t is the solution to the equation 0.8t-0.46=8(t-0.001)+1.9?

Show Answer Correct Answer: -.3266, -.3267, -49/150

The correct answer is -.3267. Applying the distributive property to the right-hand side of the given equation yields 0.8t-0.46=8t-0.008+1.9, or 0.8t-0.46=8t+1.892. Subtracting 0.8t from both sides of this equation yields -0.46=7.2t+1.892. Subtracting 1.892 from both sides of this equation yields -2.352=7.2t. Dividing both sides of this equation by 7.2 yields -2.3527.2=t. Therefore, the value of t is approximately -0.32667. Note that -.3267, -.3266, -0.326, and -0.327 are examples of ways to enter a correct answer.

Question 491 491 of 569 selected Linear Equations In 2 Variables M
x y
-6 n + 184
-3 n + 92
0 n

The table shows three values of x and their corresponding values of y , where n is a constant, for the linear relationship between x and y . What is the slope of the line that represents this relationship in the xy-plane?

  1. - 92 3

  2. - 3 92

  3. n+92-3

  4. 2n-923

Show Answer Correct Answer: A

Choice A is correct. The slope, m , of a line in the xy-plane can be found using two points on the line, (x1,y1) and (x2,y2), and the slope formula m=y2-y1x2-x1. Based on the given table, the line representing the relationship between x and y in the xy-plane passes through the points (-6,n+184), (-3,n+92), and (0,n), where n is a constant. Substituting two of these points, (-3,n+92) and (0,n), for (x1,y1) and (x2,y2), respectively, in the slope formula yields m=n-(n+92)0-(-3), which is equivalent to m=n-n-920+3, or m=-923. Therefore, the slope of the line that represents this relationship in the xy-plane is - 92 3 .

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 492 492 of 569 selected Linear Inequalities In 1 Or 2 Variables M

The minimum value of x is 12 less than 6 times another number n . Which inequality shows the possible values of x ?

  1. x6n-12

  2. x6n-12

  3. x12-6n

  4. x12-6n

Show Answer Correct Answer: B

Choice B is correct. It’s given that the minimum value of x is 12 less than 6 times another number n . Therefore, the possible values of x are all greater than or equal to the value of 12 less than 6 times n . The value of 6 times n is given by the expression 6 n . The value of 12 less than 6 n is given by the expression 6 n - 12 . Therefore, the possible values of x are all greater than or equal to 6 n - 12 . This can be shown by the inequality x6n-12.

Choice A is incorrect. This inequality shows the possible values of x if the maximum, not the minimum, value of x is 12 less than 6 times n .

Choice C is incorrect. This inequality shows the possible values of x if the maximum, not the minimum, value of x is 6 times n less than 12 , not 12 less than 6 times n .

Choice D is incorrect. This inequality shows the possible values of x if the minimum value of x is 6  times n  less than 12 , not 12 less than 6 times n .

Question 493 493 of 569 selected Linear Equations In 1 Variable M

The width of a rectangular dance floor is w feet. The length of the floor is 6 feet longer than its width. Which of the following expresses the perimeter, in feet, of the dance floor in terms of w ?

  1. 2 w plus 6

  2. 4 w plus 12

  3. w squared plus 6

  4. w squared plus 6 w

Show Answer Correct Answer: B

Choice B is correct. It is given that the width of the dance floor is w feet. The length is 6 feet longer than the width; therefore, the length of the dance floor is w plus 6. So the perimeter is w plus w, plus, open parenthesis, w plus 6, close parenthesis, plus, open parenthesis, w plus 6, close parenthesis, equals, 4 w plus 12.

Choice A is incorrect because it is the sum of one length and one width, which is only half the perimeter. Choice C is incorrect and may result from using the formula for the area instead of the formula for the perimeter and making a calculation error. Choice D is incorrect because this is the area, not the perimeter, of the dance floor.

 

Question 494 494 of 569 selected Linear Equations In 2 Variables E

x+y=350

The given equation relates the total number of maple trees, x , and the total number of birch trees, y , planted in a 14 -acre forest preserve. If 245 maple trees were planted in the forest preserve, how many birch trees were planted in the forest preserve?

  1. 14

  2. 25

  3. 105

  4. 245

Show Answer Correct Answer: C

Choice C is correct. It’s given that the equation x+y=350 relates the total number of maple trees, x, and the total number of birch trees, y, planted in a 14-acre forest preserve. It’s also given that 245 maple trees were planted in the forest preserve. Substituting 245 for x in the given equation yields 245+y=350. Subtracting 245 from both sides of this equation yields y=105. Therefore, 105 birch trees were planted in the forest preserve.

Choice A is incorrect. This is the number of acres in the forest preserve, not the number of birch trees planted in the forest preserve.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect. This is the number of maple trees planted in the forest preserve, not the number of birch trees planted in the forest preserve.

Question 495 495 of 569 selected Linear Equations In 2 Variables M

A neighborhood consists of a 2 -hectare park and a 35 -hectare residential area. The total number of trees in the neighborhood is 3,934 . The equation 2 x + 35 y = 3,934 represents this situation. Which of the following is the best interpretation of x in this context?

  1. The average number of trees per hectare in the park

  2. The average number of trees per hectare in the residential area

  3. The total number of trees in the park

  4. The total number of trees in the residential area

Show Answer Correct Answer: A

Choice A is correct. It's given that a neighborhood consists of a 2 -hectare park and a 35 -hectare residential area and that the total number of trees in the neighborhood is 3,934 . It's also given that the equation 2x+35y=3,934 represents this situation. Since the total number of trees for a given area can be determined by taking the number of hectares times the average number of trees per hectare, this must mean that the terms 2 x and 35 y correspond to the number of trees in the park and in the residential area, respectively. Since 2 x corresponds to the number of trees in the park, and 2 is the size of the park, in hectares, x must represent the average number of trees per hectare in the park.

Choice B is incorrect and may result from conceptual errors.

Choice C is incorrect and may result from conceptual errors.

Choice D is incorrect and may result from conceptual errors.

Question 496 496 of 569 selected Linear Equations In 2 Variables E

Jay walks at a speed of 3 miles per hour and runs at a speed of 5 miles per hour. He walks for w hours and runs for r hours for a combined total of 14 miles. Which equation represents this situation?

  1. 3w+5r=14

  2. 13w+15r=14

  3. 13w+15r=112

  4. 3w+5r=112

Show Answer Correct Answer: A

Choice A is correct. Since Jay walks at a speed of 3 miles per hour for w hours, Jay walks a total of 3 w miles. Since Jay runs at a speed of 5 miles per hour for r hours, Jay runs a total of 5 r miles. Therefore, the total number of miles Jay travels can be represented by 3w+5r. Since the combined total number of miles is 14 , the equation 3w+5r=14 represents this situation.

Choice B is incorrect and may result from conceptual errors.

Choice C is incorrect and may result from conceptual errors.

Choice D is incorrect and may result from conceptual errors.

Question 497 497 of 569 selected Linear Functions M

Scientists collected fallen acorns that each housed a colony of the ant species P. ohioensis and analyzed each colony's structure. For any of these colonies, if the colony has x worker ants, the equation y=0.67x+2.6, where 20x110, gives the predicted number of larvae, y , in the colony. If one of these colonies has 58 worker ants, which of the following is closest to the predicted number of larvae in the colony?

  1. 41

  2. 61

  3. 83

  4. 190

Show Answer Correct Answer: A

Choice A is correct. It's given that the equation y=0.67x+2.6, where 20x110, gives the predicted number of larvae, y , in a colony of ants if the colony has x worker ants. If one of these colonies has 58 worker ants, the predicted number of larvae in that colony can be found by substituting 58 for x in the given equation. Substituting 58 for x in the given equation yields y=0.67(58)+2.6, or y = 41.46 . Of the given choices, 41 is closest to the predicted number of larvae in the colony.

Choice B is incorrect. This is closest to the predicted number of larvae in a colony with 87 worker ants.

Choice C is incorrect. This is closest to the number of worker ants for which the predicted number of larvae in a colony is 58 .

Choice D is incorrect. This is closest to the predicted number of larvae in a colony with 280 worker ants.

Question 498 498 of 569 selected Linear Equations In 1 Variable E

A librarian has 43 books to distribute to a group of children. If he gives each child 2 books, he will have 7 books left over. How many children are in the group?

  1. 15

  2. 18

  3. 25

  4. 29

Show Answer Correct Answer: B

Choice B is correct. Subtracting the number of books left over from the total number of books results in 43 minus 7, equals 36, which is the number of books distributed. Dividing the number of books distributed by the number of books given to each child results in the fraction 36 over 2, equals 18.

Choice A is incorrect and results from dividing the total number of books by the number of books given to each child, the fraction 43 over 2, is approximately equal to 22, then subtracting the number of books left over from the result, 22 minus 7, equals 15. Choice C is incorrect and results from adding the number of books left over to the total number of books, 43 plus 7, equals 50, then dividing the result by the number of books given to each child, the fraction 50 over 2, equals 25. Choice D is incorrect and results from dividing the total number of books by the number of books given to each child, the fraction 43 over 2, is approximately equal to 22, then adding the number of books left over, 22 plus 7, equals 29.

Question 499 499 of 569 selected Linear Equations In 1 Variable E

A manager is responsible for ordering supplies for a shaved ice shop. The shop's inventory starts with 4,500 paper cups, and the manager estimates that 70 of these paper cups are used each day. Based on this estimate, in how many days will the supply of paper cups reach 1,700 ?

  1. 20

  2. 40

  3. 60

  4. 80

Show Answer Correct Answer: B

Choice B is correct. It’s given that the shop’s inventory starts with 4,500 paper cups and that the manager estimates that 70 of these paper cups are used each day. Let x represent the number of days in which the estimated supply of paper cups will reach 1,700. The equation 4,500-70x=1,700 represents this situation. Subtracting 4,500 from both sides of this equation yields -70x=-2,800. Dividing both sides of this equation by -70 yields x = 40 . Therefore, based on this estimate, the supply of paper cups will reach 1,700 in 40 days.

Choice A is incorrect. After 20 days, the estimated supply of paper cups would be 4,500-70(20), or 3,100 cups, not 1,700 cups.

Choice C is incorrect. After 60 days, the estimated supply of paper cups would be 4,500-70(60), or 300 cups, not 1,700 cups.

Choice D is incorrect. After 80 days, the estimated supply of paper cups would be 4,500-70(80), or -1,100 cups, which isn't possible.

Question 500 500 of 569 selected Linear Functions E

The graph of the function f is a line in the xy-plane. If the line has slope three-fourths and f of 0, equals 3, which of the following defines f?

  1. f of x equals, three fourths x, minus 3

  2. f of x equals, three fourths x, plus 3

  3. f of x equals, 4 x, minus 3

  4. f of x equals, 4 x, plus 3

Show Answer Correct Answer: B

Choice B is correct. The equation for the function f in the xy-plane can be represented by f of x equals, m x plus b, where m is the slope and b is the y-coordinate of the y-intercept. Since it’s given that the line has a slope of three fourths, it follows that m equals three fourths in f of x equals, m x plus b, which yields y equals, three fourths x, plus b. It’s given that f of 0 equals 3. This implies that the graph of the function f in the xy-plane passes through the point with coordinates 0 comma 3. Thus, the y-coordinate of the y-intercept of the graph is 3, so b equals 3 in f of x equals, three fourths x, plus b, which yields f of x equals, three fourths x, plus 3. Therefore, the equation f of x equals, three fourths x, plus 3 defines the function f.

Choice A is incorrect and may result from a sign error for the y-intercept. Choice C is incorrect and may result from using the denominator of the given slope as m in f of x equals, m x plus b, in addition to a sign error for the y-intercept. Choice D is incorrect and may result from using the denominator of the given slope as m in f of x equals, m x plus b.

 

Question 501 501 of 569 selected Systems Of 2 Linear Equations In 2 Variables M

8x+y=5

y=9x+1 

The solution to the given system of equations is (x,y). What is the value of x

  1. -6

  2. 4 17

  3. 6 17

  4. 4

Show Answer Correct Answer: B

Choice B is correct. The second equation in the given system is y = 9 x + 1 . Substituting 9 x + 1 for y in the first equation in the given system yields 8x+9x+1=5, which is equivalent to 17 x + 1 = 5 . Subtracting 1 from both sides of this equation yields 17 x = 4 . Dividing both sides of this equation by 17 yields x = 4 17 .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 502 502 of 569 selected Linear Functions M

  • The line begins on the y axis above the x axis.
  • The line slants sharply up from left to right.

The graph represents the total charge, in dollars, by an electrician for x hours of work. The electrician charges a onetime fee plus an hourly rate. What is the best interpretation of the slope of the graph?

  1. The electrician’s hourly rate

  2. The electrician’s onetime fee

  3. The maximum amount that the electrician charges

  4. The total amount that the electrician charges

Show Answer Correct Answer: A

Choice A is correct. It’s given that the electrician charges a onetime fee plus an hourly rate. It's also given that the graph represents the total charge, in dollars, for x hours of work. This graph shows a linear relationship in the xy-plane. Thus, the total charge y , in dollars, for x hours of work can be represented as y=mx+b, where m is the slope and (0,b) is the y-intercept of the graph of the equation in the xy-plane. Since the given graph represents the total charge, in dollars, by an electrician for x hours of work, it follows that its slope is m , or the electrician’s hourly rate.

Choice B is incorrect. The electrician's onetime fee is represented by the y-coordinate of the y-intercept, not the slope, of the graph.

Choice C is incorrect and may result from conceptual errors.

Choice D is incorrect and may result from conceptual errors.

Question 503 503 of 569 selected Linear Equations In 2 Variables E

In 2010 , a swim club had a total of 35 swimmers, each classified as either advanced or intermediate. From 2010 to 2020 , the number of advanced swimmers in the club increased by approximately 53 %, and the number of intermediate swimmers in the club increased by approximately 44 %. The total number of swimmers in the club increased by approximately 49 %. Which equation best represents this situation, where a represents the number of advanced swimmers in the club in 2010 and b represents the number of intermediate swimmers in the club in 2010 ?

  1. 1.53 a + 1.49 b =35(1.44)

  2. 1.49 a + 0.53 b =35(1.44)

  3. 1.53 a + 1.44 b =35(1.49)

  4. 1.44 a + 1.53 b =35(1.49)

Show Answer Correct Answer: C

Choice C is correct. It’s given that in 2010, a swim club had a total of 35 swimmers, each classified as either advanced or intermediate, and that a represents the number of advanced swimmers in 2010 and b represents the number of intermediate swimmers in 2010. It's also given that from 2010 to 2020, the number of advanced swimmers in the club increased by approximately 53% and the number of intermediate swimmers in the club increased by approximately 44%. Thus, in 2020, the approximate number of advanced swimmers in the club can be represented as 1.53a and the approximate number of intermediate swimmers in the club can be represented as 1.44b. It's given that the total number of swimmers in the club increased by approximately 49% from 2010 to 2020. Since the club had 35 swimmers in 2010, it follows that the total number of swimmers in 2020 can be represented as 35(1.49). Since the sum of the number of advanced swimmers in 2020 and the number of intermediate swimmers in 2020 equals the total number of swimmers in 2020, the equation 1.53a+1.44b=35(1.49) best represents this situation.

Choice A is incorrect. This equation represents a situation where the number of intermediate swimmers in the club in 2020 increased by approximately 49%, not 44%, and the total number of swimmers in the club in 2020 increased by approximately 44%, not 49%.

Choice B is incorrect and may result from conceptual errors.

Choice D is incorrect. This equation represents a situation where the number of advanced swimmers in the club in 2020 increased by approximately 44%, not 53%, and the number of intermediate swimmers in the club in 2020 increased by approximately 53%, not 44%.

Question 504 504 of 569 selected Systems Of 2 Linear Equations In 2 Variables H

y = 6 x + 18

One of the equations in a system of two linear equations is given. The system has no solution. Which equation could be the second equation in the system?

  1. - 6 x + y = 18

  2. - 6 x + y = 22

  3. - 12 x + y = 36

  4. - 12 x + y = 18

Show Answer Correct Answer: B

Choice B is correct. A system of two linear equations in two variables, x and y , has no solution if the lines represented by the equations in the xy-plane are parallel and distinct. Lines represented by equations in standard form, Ax+By=C and Dx+Ey=F, are parallel if the coefficients for x and y in one equation are proportional to the corresponding coefficients in the other equation, meaning DA=EB; and the lines are distinct if the constants are not proportional, meaning FC is not equal to DA or EB. The given equation, y=6x+18, can be written in standard form by subtracting 6x from both sides of the equation to yield -6x+y=18. Therefore, the given equation can be written in the form Ax+By=C, where A=-6, B=1, and C=18. The equation in choice B, -6x+y=22, is written in the form Dx+Ey=F, where D=-6, E=1, and F=22. Therefore,  DA=-6-6, which can be rewritten as DA=1; EB=11, which can be rewritten as EB=1; and FC=2218, which can be rewritten as FC=119. Since DA=1EB=1, and FC is not equal to 1 , it follows that the given equation and the equation -6x+y=22 are parallel and distinct. Therefore, a system of two linear equations consisting of the given equation and the equation -6x+y=22 has no solution. Thus, the equation in choice B could be the second equation in the system.

Choice A is incorrect. The equation -6x+y=18 and the given equation represent the same line in the xy-plane. Therefore, a system of these linear equations would have infinitely many solutions, rather than no solution. 

Choice C is incorrect. The equation -12x+y=36 and the given equation represent lines in the xy-plane that are distinct and not parallel. Therefore, a system of these linear equations would have exactly one solution, rather than no solution.

Choice D is incorrect. The equation -12x+y=18 and the given equation represent lines in the xy-plane that are distinct and not parallel. Therefore, a system of these linear equations would have exactly one solution, rather than no solution.

Question 505 505 of 569 selected Linear Functions E

Oxygen gas is placed inside a tank with a constant volume. The graph shows the estimated temperature y , in kelvins, of the oxygen gas when its pressure is x atmospheres.

  • The line slants sharply up from left to right.
  • The line begins at the approximate point (1.6 comma 190).
  • The line passes through the following approximate points:
    • (3 comma 350)
    • (6 comma 700)
    • (8 comma 933)

What is the estimated temperature, in kelvins, of the oxygen gas when its pressure is 6 atmospheres?

  1. 6

  2. 60

  3. 700

  4. 760

Show Answer Correct Answer: C

Choice C is correct. For the graph shown, the x-axis represents pressure, in atmospheres, and the y-axis represents temperature, in kelvins. Therefore, the estimated temperature, in kelvins, of the oxygen gas when its pressure is 6 atmospheres is represented by the y-coordinate of the point on the graph that has an x-coordinate of 6 . The point on the graph with an x-coordinate of 6 has a y-coordinate of approximately 700 . Therefore, the estimated temperature, in kelvins, of the oxygen gas when its pressure is 6 atmospheres is 700 .

Choice A is incorrect. This is the pressure, in atmospheres, not the estimated temperature, in kelvins, of the oxygen gas when its pressure is 6 atmospheres.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 506 506 of 569 selected Systems Of 2 Linear Equations In 2 Variables E

An online bookstore sells novels and magazines. Each novel sells for $4, and each magazine sells for $1. If Sadie purchased a total of 11 novels and magazines that have a combined selling price of $20, how many novels did she purchase?

  1. 2

  2. 3

  3. 4

  4. 5

Show Answer Correct Answer: B

Choice B is correct. Let n be the number of novels and m be the number of magazines that Sadie purchased. If Sadie purchased a total of 11 novels and magazines, then n plus m, equals 11. It is given that the combined price of 11 novels and magazines is $20. Since each novel sells for $4 and each magazine sells for $1, it follows that 4 n plus m, equals 20. So the system of equations below must hold.

4 n plus m, equals 20; n plus m, equals 11

Subtracting corresponding sides of the second equation from the first equation yields 3 n equals 9, so n equals 3. Therefore, Sadie purchased 3 novels.

Choice A is incorrect. If 2 novels were purchased, then a total of $8 was spent on novels. That leaves $12 to be spent on magazines, which means that 12 magazines would have been purchased. However, Sadie purchased a total of 11 novels and magazines. Choices C and D are incorrect. If 4 novels were purchased, then a total of $16 was spent on novels. That leaves $4 to be spent on magazines, which means that 4 magazines would have been purchased. By the same logic, if Sadie purchased 5 novels, she would have no money at all ($0) to buy magazines. However, Sadie purchased a total of 11 novels and magazines.

 

Question 507 507 of 569 selected Linear Equations In 1 Variable E

6 x plus k, equals, 6 x plus 5

In the given equation, k is a constant. If the equation has infinitely many solutions, what is the value of k ?

Show Answer

The correct answer is 5. Subtracting 6 x from both sides of the given equation gives k equals 5, so for any value of x, 6 x plus k, equals, 6 x plus 5 if and only if k equals 5. Therefore, if the given equation has infinitely many solutions, the value of k is 5.

Question 508 508 of 569 selected Linear Equations In 2 Variables H

What is the y-coordinate of the y-intercept of the graph of 3 x 7 = - 5 y 9 + 21 in the xy-plane?

Show Answer Correct Answer: 189/5, 37.8

The correct answer is 189 5 . A y-intercept of a graph in the xy-plane is a point where the graph intersects the y-axis, which is a point with an x-coordinate of 0 . Substituting 0 for x in the given equation yields 3(0)7=-5y9+21, or 0=-5y9+21. Subtracting 21 from both sides of this equation yields -21=-5y9. Multiplying both sides of this equation by -9 yields 189 = 5 y . Dividing both sides of this equation by 5 yields 189 5 = y . Therefore, the y-coordinate of the y-intercept of the graph of the given equation in the xy-plane is 189 5 . Note that 189/5 and 37.8 are examples of ways to enter a correct answer.

Question 509 509 of 569 selected Linear Equations In 1 Variable H

If x-57=x-59, the value of x-5 is between which of the following pairs of values?

  1. -9 and -7

  2. -3 and 3

  3. 4.5 and 5.5

  4. 6.75 and 9.25

Show Answer Correct Answer: B

Choice B is correct. Multiplying both sides of the given equation by (7)(9), or 63, yields (63)(x-57)=(63)(x-59), or 9(x-5)=7(x-5). Subtracting 7(x-5) from both sides of this equation yields 2(x-5)=0. Dividing both sides of this equation by 2 yields x-5=0. Therefore, if x-57=x-59, then the value of x-5 is 0. It follows that of the given choices, the value of x-5 is between -3 and 3.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 510 510 of 569 selected Linear Equations In 2 Variables H

How many liters of a 25% saline solution must be added to 3 liters of a 10% saline solution to obtain a 15% saline solution?

Show Answer

The correct answer is 1.5. The total amount, in liters, of a saline solution can be expressed as the liters of each type of saline solution multiplied by the percent concentration of the saline solution. This gives 3 times 0 point 1 0, x times 0 point 2 5, and open parenthesis, x plus 3, close parenthesis, times, open parenthesis, 0 point 1 5, close parenthesis, where x is the amount, in liters, of 25% saline solution and 10%, 15%, and 25% are represented as 0.10, 0.15, and 0.25, respectively. Thus, the equation 3 times 0 point 1 0, plus 0 point 2 5 x, equals, 0 point 1 5 times, open parenthesis, x plus 3, close parenthesis must be true. Multiplying 3 by 0.10 and distributing 0.15 to open parenthesis, x plus 3, close parenthesis yields 0 point 3 0, plus 0 point 2 5 x, equals, 0 point 1 5 x, plus 0 point 4 5. Subtracting 0.15x and 0.30 from each side of the equation gives 0 point 1 0 x, equals 0 point 1 5. Dividing each side of the equation by 0.10 yields x equals 1 point 5. Note that 1.5 and 3/2 are examples of ways to enter a correct answer.

Question 511 511 of 569 selected Systems Of 2 Linear Equations In 2 Variables E

  • For the first line in the system:
    • The line slants gradually down from left to right.
    • The line passes through the following points:
      • (negative 7 comma 0)
      • (negative 4 comma negative 3)
      • (0 comma negative 7)
  • For the second line in the system:
    • The line slants gradually up from left to right.
    • The line passes through the following points:
      • (negative 5 comma negative 4)
      • (negative 4 comma negative 3)
      • (negative 3 comma negative 2)

The graph of a system of linear equations is shown. What is the solution (x,y) to the system?

  1. (0,-7)

  2. (0,-3)

  3. (-4,-3)

  4. (-4,0)

Show Answer Correct Answer: C

Choice C is correct. The solution to a system of linear equations is represented by the point that lies on the graph of each equation in the system, or the point where the lines intersect on a graph. On the graph shown, the two lines intersect at the point (-4,-3). Therefore, the solution to the system is (-4,-3).

Choice A is incorrect. This is the y-intercept of the graph of one of the lines shown, not the intersection point of the two lines.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 512 512 of 569 selected Systems Of 2 Linear Equations In 2 Variables M

Equation 1: 5 x plus 3 y, equals 38; Equation 2: x plus 3 y, equals 10

In the solution x comma y to the system of equations above, what is the value of x ?

Show Answer

The correct answer is 7. Subtracting the second equation from the first equation eliminates the variable y.

Equation 1, 5 x, plus 3 y, equals 38; Equation 2,  x plus 3 y, equals 10; Subtracting equation 2 from equation 1 yields equation 3, which is 4 x, equals 28.

Dividing both sides of the resulting equation by 4 yields x = 7.

Question 513 513 of 569 selected Linear Equations In 1 Variable H

A science teacher is preparing the 5 stations of a science laboratory. Each station will have either Experiment A materials or Experiment B materials, but not both. Experiment A requires 6 teaspoons of salt, and Experiment B requires 4 teaspoons of salt. If x is the number of stations that will be set up for Experiment A and the remaining stations will be set up for Experiment B, which of the following expressions represents the total number of teaspoons of salt required?

  1. 5 x
  2. 10 x
  3. 2 x plus 20
  4. 10 x plus 20
Show Answer Correct Answer: C

Choice C is correct. It is given that x represents the number of stations that will be set up for Experiment A and that there will be 5 stations total, so it follows that 5 – x is the number of stations that will be set up for Experiment B. It is also given that Experiment A requires 6 teaspoons of salt and that Experiment B requires 4 teaspoons of salt, so the total number of teaspoons of salt required is 6x + 4(5 – x), which simplifies to 2x + 20.

Choices A, B, and D are incorrect and may be the result of not understanding the description of the context.

Question 514 514 of 569 selected Linear Functions E

The function f is defined by f(x)=-3x+60. What is the value of f(x) when x=-8?

  1. 49

  2. 52

  3. 57

  4. 84

Show Answer Correct Answer: D

Choice D is correct. The value of f(x) when x = - 8 can be found by substituting - 8 for x in the given function. This yields f(-8)=-3(-8)+60, or f(-8)=84. Therefore, when x = - 8 , the value of f(x) is 84 .

Choice A is incorrect. This is the value of (-3+(-8))+60, not -3(-8)+60.

Choice B is incorrect. This is the value of -8+60, not -3(-8)+60.

Choice C is incorrect. This is the value of -3+60, not -3(-8)+60.

Question 515 515 of 569 selected Linear Equations In 2 Variables M

2 x + y = 37

In triangle Q R S , sides Q R and R S each have a length of x centimeters and side SQ has a length of y centimeters. The given equation represents this situation. Which of the following is the best interpretation of 37 in this context?

  1. The difference, in centimeters, between the lengths of sides Q R and SQ

  2. The difference, in centimeters, between the lengths of sides Q R and R S

  3. The sum of the lengths, in centimeters, of the three sides of the triangle

  4. The length, in centimeters, of one of the two sides of equal length

Show Answer Correct Answer: C

Choice C is correct. It's given that in triangle QRS, sides QR and RS each have a length of x centimeters. Therefore, the expression 2 x represents the sum of the lengths, in centimeters, of sides QR and RS. It's also given that side SQ has a length of y centimeters. Therefore, the expression 2x+y represents the sum of the lengths, in centimeters, of sides QR, RS, and SQ. Since 2x+y is the sum of the lengths, in centimeters, of the three sides of the triangle and 2x+y=37, it follows that 37 is the sum of the lengths, in centimeters, of the three sides of the triangle.

Choice A is incorrect. The difference, in centimeters, between the lengths of sides QR and SQ is x - y , not 37 .

Choice B is incorrect. The difference, in centimeters, between the lengths of sides QR and RS is x-x, or 0 , not 37 .

Choice D is incorrect. The length, in centimeters, of one of the two sides of equal length is x , not 37 .

Question 516 516 of 569 selected Linear Equations In 2 Variables M

What is the slope of the graph of y=14(27x+15)+7x in the xy-plane?

Show Answer Correct Answer: 13.75, 55/4

The correct answer is 554. In the xy-plane, the graph of an equation in the form y = m x + b , where m and b are constants, has a slope of m and a y-intercept of (0,b). Applying the distributive property to the right-hand side of the given equation yields y=274x+154+7x. Combining like terms yields y=554x+154. This equation is in the form y = m x + b , where m = 55 4 and b = 15 4 . It follows that the slope of the graph of y=14(27x+15)+7x in the xy-plane is 554. Note that 55/4 and 13.75 are examples of ways to enter a correct answer.

Question 517 517 of 569 selected Linear Equations In 1 Variable M

If 2 times, open parenthesis, x minus 5, close parenthesis, plus 3 times, open parenthesis, x minus 5, close parenthesis, equals 10, what is the value of x minus 5 ?

  1. 2

  2. 5

  3. 7

  4. 12

Show Answer Correct Answer: A

Choice A is correct. Adding the like terms on the left-hand side of the given equation yields 5 times, open parenthesis, x minus 5, close parenthesis, equals 10. Dividing both sides of this equation by 5 yields x minus 5, equals 2.

Choice B is incorrect and may result from subtracting 5, not dividing by 5, on both sides of the equation 5 times, open parenthesis, x minus 5, close parenthesis, equals 10. Choice C is incorrect. This is the value of x, not the value of x minus 5. Choice D is incorrect. This is the value of x plus 5, not the value of x minus 5.

Question 518 518 of 569 selected Linear Functions M

The relationship between two variables, x and y , is linear. For every increase in the value of x by 1 , the value of y increases by 8 . When the value of x is 2 , the value of y is 18 . Which equation represents this relationship?

  1. y = 2 x + 18

  2. y = 2 x + 8

  3. y = 8 x + 2

  4. y = 3 x + 26

Show Answer Correct Answer: C

Choice C is correct. It’s given that the relationship between x and y is linear. An equation representing a linear relationship can be written in the form y = m x + b , where m is the slope and b is the y-coordinate of the y-intercept of the graph of the relationship in the xy-plane. It’s given that for every increase in the value of x by 1 , the value of y increases by 8 . The slope of a line can be expressed as the change in y over the change in x . Thus, the slope, m , of the line representing this relationship can be expressed as 81, or 8 . Substituting 8 for m in the equation y = m x + b yields y = 8 x + b . It's also given that when the value of x is 2 , the value of y is 18 . Substituting 2 for x and 18 for y in the equation y = 8 x + b yields 18=8(2)+b, or 18=16+b. Subtracting 16 from each side of this equation yields 2 = b . Substituting 2 for b in the equation y = 8 x + b yields y = 8 x + 2 . Therefore, the equation y = 8 x + 2 represents this relationship.

Choice A is incorrect. This equation represents a relationship where for every increase in the value of x by 1 , the value of y increases by 2 , not 8 , and when the value of x is 2 , the value of y is 22 , not 18 .

Choice B is incorrect. This equation represents a relationship where for every increase in the value of x by 1 , the value of y increases by 2 , not 8 , and when the value of x is 2 , the value of y is 12 , not 18 .

Choice D is incorrect. This equation represents a relationship where for every increase in the value of x by 1 , the value of y increases by 3 , not 8 , and when the value of x is 2 , the value of y is 32 , not 18 .

Question 519 519 of 569 selected Linear Inequalities In 1 Or 2 Variables E

2l+2w27

A rectangle has length l and width w . The inequality gives the possible values of l and w for which the perimeter of this rectangle is less than or equal to 27 . Which statement is the best interpretation of (l,w)=(8,3) in this context?

  1. If the rectangle has length 3 and width 8 , its perimeter is less than or equal to 27 .

  2. If the rectangle has length 8 and width 3 , its perimeter is less than or equal to 27 .

  3. If the rectangle has length 3 and width 8 , its perimeter is greater than or equal to 27 .

  4. If the rectangle has length 8 and width 3 , its perimeter is greater than or equal to 27 .

Show Answer Correct Answer: B

Choice B is correct. It’s given that a rectangle has length l and width w, and the inequality 2l+2w27 gives the possible values of l and w for which the perimeter of this rectangle is less than or equal to 27. To determine the best interpretation of (l,w)=(8,3) in this context, the values can be substituted in the given inequality. Substituting 8 for l and 3 for w in this inequality yields 2(8)+2(3)27, which is equivalent to 16+627, or 2227. Since this inequality is true, if the rectangle has length 8 and width 3, its perimeter is less than or equal to 27.

Choice A is incorrect. The interpretation of (l,w)=(8,3) implies that the rectangle has length 8 and width 3, not length 3 and width 8.

Choice C is incorrect. The interpretation of (l,w)=(8,3) implies that the rectangle has length 8 and width 3, not length 3 and width 8.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 520 520 of 569 selected Linear Inequalities In 1 Or 2 Variables E

Maria plans to rent a boat. The boat rental costs $60 per hour, and she will also have to pay for a water safety course that costs $10. Maria wants to spend no more than $280 for the rental and the course. If the boat rental is available only for a whole number of hours, what is the maximum number of hours for which Maria can rent the boat?

Show Answer

The correct answer is 4. The equation 60 h plus 10, is less than or equal to 280, where h is the number of hours the boat has been rented, can be written to represent the situation. Subtracting 10 from both sides and then dividing by 60 yields h is less than or equal to 4 point 5. Since the boat can be rented only for whole numbers of hours, the maximum number of hours for which Maria can rent the boat is 4.

Question 521 521 of 569 selected Linear Functions E

Robert rented a truck to transport materials he purchased from a hardware store. He was charged an initial fee of $20.00 plus an additional $0.70 per mile driven. If the truck was driven 38 miles, what was the total amount Robert was charged?

  1. $46.60

  2. $52.90

  3. $66.90

  4. $86.50

Show Answer Correct Answer: A

Choice A is correct. It’s given that Robert was charged an initial fee of $20.00 to rent the truck plus an additional $0.70 per mile driven. Let m represent the number of miles the truck was driven. Since the rental charge is $0.70 per mile driven, 0 point 7 0 m represents the amount Robert was charged for m miles driven. Let c equal the total amount, in dollars, Robert was charged to rent the truck. The total amount can be represented by the equation c equals, 20 point 0 0 plus 0 point 7 0 m. It’s given that the truck was driven 38 miles, thus m equals 38. Substituting 38 into the equation gives c equals, 20 point 0 0 plus 0 point 7 0, times 38. Multiplying 0 point 7 0 times 38 gives c equals, 20 point 0 0 plus 26 point 6 0. Adding these values gives c equals 46 point 6 0, so the total amount Robert was charged is $46.60.

Choices B, C, and D are incorrect and may result from setting up the equation incorrectly or from making calculation errors.

 

Question 522 522 of 569 selected Linear Equations In 2 Variables M

A line passes through the points (4,6) and (15,24) in the xy-plane. What is the slope of the line?

Show Answer Correct Answer: 1.636, 18/11

The correct answer is 18 11 . For a line that passes through the points (x1,y1) and (x2,y2) in the xy-plane, the slope of the line can be calculated using the slope formula, m=y2-y1x2-x1. It's given that a line passes through the points (4,6) and (15,24) in the xy-plane. Substituting (4,6) for (x1,y1) and (15,24) for (x2,y2) in the slope formula, m=y2-y1x2-x1, yields m=24-615-4, or m=1811. Therefore, the slope of the line is 18 11 . Note that 18/11 and 1.636 are examples of ways to enter a correct answer.

Question 523 523 of 569 selected Linear Equations In 1 Variable M

14(x+5)-13(x+5)=-7

What value of x  is the solution to the given equation?

  1. -12

  2. -5

  3. 79

  4. 204

Show Answer Correct Answer: C

Choice C is correct. For the given equation, (x+5) is a factor of both terms on the left-hand side. Therefore, the given equation can be rewritten as (14-13)(x+5)=-7, or (312-412)(x+5)=-7, which is equivalent to -112(x+5)=-7. Multiplying both sides of this equation by -12 yields x+5=84. Subtracting 5 from both sides of this equation yields x=79

Choice A is incorrect. This is the value of x for which the left-hand side of the given equation equals 7 12 , not -7 .

Choice B is incorrect. This is the value of x for which the left-hand side of the given equation equals 0 , not -7 .

Choice D is incorrect. This is the value of x for which the left-hand side of the given equation equals - 209 12 , not -7 .

Question 524 524 of 569 selected Systems Of 2 Linear Equations In 2 Variables E

A discount airline sells a certain number of tickets, x, for a flight for $90 each. It sells the number of remaining tickets, y, for $250 each. For a particular flight, the airline sold 120 tickets and collected a total of $27,600 from the sale of those tickets. Which system of equations represents this relationship between x and y ?

  1. open brace, x plus y equals 120, and, 90 x plus 250 y equals 27,600

  2. open brace, x plus y equals 120, and, 90 x plus 250 y equals 120 times 27,600

  3. open brace, x plus y equals 27,600, and, 90 x plus 250 y equals 120 times 27,600

  4. open brace, 90 x equals 250 y, and, 120 x plus 120 y equals 27,600

Show Answer Correct Answer: A

Choice A is correct. The airline sold two types of tickets for this flight: x tickets at $90 each and the remaining  tickets, y, at $250 each. Because the airline sold a total of 120 tickets for this flight, it must be true that x + y = 120. The amount, in dollars, collected from the sale of x tickets at $90 each is represented by 90x. The amount, in dollars, collected from the sale of the remaining y tickets at $250 each is represented by 250y. It is given that a total of $27,600 was collected from the sale of all tickets. Therefore, it must also be true that 90x + 250y = 27,600.

Choice B is incorrect. The total number of tickets sold is represented correctly as x + y = 120. The total amount, in dollars, collected from the sale of the x tickets at $90 each and the remaining tickets, y, at $250 has been correctly represented as 90x + 250y. However, according to the information given, this total should be equal to 27,600, not 120(27,600) dollars. Choice C is incorrect. The total number of tickets sold has been correctly represented as x + y. However, according to the information given, this total should be equal to 120, not 27,600, as shown in choice C. The total amount, in dollars, collected from the sale of the x tickets at $90 each and the remaining tickets, y, at $250 has been correctly represented as 90x + 250y. However, according to the information given, this total should be equal to 27,600, not 120(27,600) dollars. Choice D is incorrect. The two equations given in choice D have no meaning in this context.

Question 525 525 of 569 selected Linear Functions M

f(x)=2x+3

For the given function f , the graph of y=f(x) in the xy-plane is parallel to line j . What is the slope of line j ?

Show Answer Correct Answer: 2

The correct answer is 2 . It’s given that function f is defined by f(x)=2x+3. Therefore, the equation representing the graph of y=f(x) in the xy-plane is y=2x+3, and the graph is a line. For a linear equation in the form y=mx+b, m represents the slope of the line. Since the value of m in the equation y=2x+3 is 2 , the slope of the line defined by function f is 2 . It's given that line j is parallel to the line defined by function f . The slopes of parallel lines are equal. Therefore, the slope of line j is also 2 .

Question 526 526 of 569 selected Linear Functions H

The equation h=9(v-273.15)5+32 gives the corresponding temperature h , in degrees Fahrenheit, of any substance that has a temperature of v kelvins, where v>0. If a substance has a temperature of 467.33 degrees Fahrenheit, what is the corresponding temperature, in kelvins, of this substance?

Show Answer Correct Answer: 515

The correct answer is 515. It’s given that the equation h=9(v273.15)5+32 gives the corresponding temperature h, in degrees Fahrenheit, of any substance that has a temperature of v kelvins, where v>0. Substituting 467.33 for h in the given equation yields 467.33=9(v273.15)5+32. Subtracting 32 from both sides of this equation yields 435.33=9(v273.15)5. Multiplying both sides of this equation by 5 yields 2,176.65=9(v273.15). Dividing both sides of this equation by 9 yields 241.85=v273.15. Adding 273.15 to both sides of this equation yields 515=v. Therefore, if a substance has a temperature of 467.33 degrees Fahrenheit, the corresponding temperature, in kelvins, of this substance is 515.

Question 527 527 of 569 selected Linear Equations In 2 Variables E

  • The line slants gradually up from left to right.
  • The line passes through the following points:
    • (0 comma 8)
    • (5 comma 9)
    • (10 comma 10)

What is the y-intercept of the line graphed?

  1. (0,-8)

  2. (0,-18)

  3. (0,0)

  4. (0,8)

Show Answer Correct Answer: D

Choice D is correct. The y-intercept of a line graphed in the xy-plane is the point where the line intersects the y-axis. The line graphed intersects the y-axis at the point (0,8). Therefore, the y-intercept of the line graphed is (0,8).

Choice A is incorrect and may result from conceptual errors.

Choice B is incorrect and may result from conceptual errors.

Choice C is incorrect and may result from conceptual errors.

Question 528 528 of 569 selected Linear Equations In 2 Variables E

The equation 46 = 2 a + 2 b gives the relationship between the side lengths a and b of a certain parallelogram. If a = 9 , what is the value of b ?

Show Answer Correct Answer: 14

The correct answer is 14 . It's given that the equation 46=2a+2b gives the relationship between the side lengths a and b of a certain parallelogram. Substituting 9 for a in the given equation yields 46=2(9)+2b, or 46=18+2b. Subtracting 18 from both sides of this equation yields 28=2b. Dividing both sides of this equation by 2 yields 14=b. Therefore, if a=9, the value of b is 14 .

Question 529 529 of 569 selected Systems Of 2 Linear Equations In 2 Variables E

A movie theater charges $11 for each full-price ticket and $8.25 for each reduced-price ticket. For one movie showing, the theater sold a total of 214 full-price and reduced-price tickets for $2,145. Which of the following systems of equations could be used to determine the number of full-price tickets, f, and the number of reduced-price tickets, r, sold?

  1. f plus r, equals 2,145, and, 11 f plus 8 point 2 5 r, equals 214
  2. f plus r, equals 214, and, 11 f plus 8 point 2 5 r, equals 2,145
  3. f plus r, equals 214, and, 8 point 2 5 f plus 11 r, equals 2,145
  4. f plus r, equals 2,145, and, 8 point 2 5 f plus 11 r, equals 214
Show Answer Correct Answer: B

Choice B is correct. The movie theater sells f full-price tickets and r reduced-price tickets, so the total number of tickets sold is f + r. Since the movie theater sold a total of 214 full-price and reduced-price tickets for one movie showing, it follows that f + r = 214. The movie theater charges $11 for each full-price ticket; thus, the sales for full-price tickets, in dollars, is given by 11f. The movie theater charges $8.25 for each reduced-price ticket; thus, the sales for reduced-price tickets, in dollars, is given by 8.25r. Therefore, the total sales, in dollars, for the movie showing is given by 11f + 8.25r. Since the total sales for all full-price and reduced-price tickets is $2,145, it follows that 11f + 8.25r = 2,145.

Choice A is incorrect. This system of equations suggests that the movie theater sold a total of 2,145 full-price and reduced-price tickets for a total of $214. Choice C is incorrect. This system suggests that the movie theater charges $8.25 for each full-price ticket and $11 for each reduced-price ticket. Choice D is incorrect. This system suggests that the movie theater charges $8.25 for each full-price ticket and $11 for each reduced-price ticket and sold a total of 2,145 tickets for a total of $214.

Question 530 530 of 569 selected Linear Functions M

The function f is defined by f of x equals, m x plus b, where m and b are constants. If f of 0 equals 18 and f of 1 equals 20, what is the value of m ?

Show Answer

The correct answer is 2. The slope-intercept form of an equation for a line is y equals, m x plus b, where m is the slope and b is the y-coordinate of the y-intercept. Two ordered pairs, x sub 1 comma y sub 1 and x sub 2 comma y sub 2, can be used to compute the slope using the formula m equals, the fraction with numerator y sub 2, minus y sub 1, and denominator x sub 2, minus x sub 1, end fraction. It’s given that f of 0 equals 18 and f of 1 equals 20 ; therefore, the two ordered pairs for this line are 0 comma 18 and 1 comma 20. Substituting these values for x sub 1 comma y sub 1 and x sub 2 comma y sub 2 gives the fraction with numerator 20 minus 18, and denominator 1 minus 0, end fraction, equals, the fraction 2 over 1, or 2.

Question 531 531 of 569 selected Linear Equations In 2 Variables M

A line in the xy-plane has a slope of 9 and passes through the point (0,-5). The equation y = p x + r defines the line, where p and r are constants. What is the value of p ?

Show Answer Correct Answer: 9

The correct answer is 9 . It’s given that the equation y=px+r defines the line. In this equation, p represents the slope of the line and r represents the y-coordinate of the y-intercept of the line. It’s given that the line has a slope of 9 . Therefore, the value of p is 9 .

Question 532 532 of 569 selected Linear Functions E

P of t equals, 250 plus 10 t

The population of snow leopards in a certain area can be modeled by the function P defined above, where P of t is the population t years after 1990. Of the following, which is the best interpretation of the equation P of 30 equals 550 ?

  1. The snow leopard population in this area is predicted to be 30 in the year 2020.

  2. The snow leopard population in this area is predicted to be 30 in the year 2030.

  3. The snow leopard population in this area is predicted to be 550 in the year 2020.

  4. The snow leopard population in this area is predicted to be 550 in the year 2030.

Show Answer Correct Answer: C

Choice C is correct. It’s given that P of t represents the population of snow leopards t years after 1990. P of 30 equals 550 corresponds to t equals 30 and P of t equals 550. It follows that t equals 30 corresponds to 30 years after 1990, or 2020. Thus, the best interpretation of P of 30 equals 550 is that the snow leopard population in this area is predicted to be 550 in the year 2020.

Choices A and B are incorrect and may result from reversing the interpretations of t and P of t. Choice D is incorrect and may result from determining that 30 years after 1990 is 2030, not 2020.

 

Question 533 533 of 569 selected Linear Functions H

For groups of 25 or more people, a museum charges $21 per person for the first 25 people and $14 for each additional person. Which function f gives the total charge, in dollars, for a tour group with n people, where n25?

  1. f(n)= 14 n + 175

  2. f(n)= 14 n + 525

  3. f(n)= 35 n - 350

  4. f(n)= 14 n + 21

Show Answer Correct Answer: A

Choice A is correct. A tour group with n people, where n25, can be split into two subgroups: the first 25 people and the additional n - 25 people. Since the museum charges $21 per person for the first 25 people and $14 for each additional person, the charge for the first 25 people is $21(25) and the charge for the additional n - 25 people is $14(n-25). Therefore, the total charge, in dollars, is given by the function f(n)=21(25)+14(n-25), or f(n)=14n+175.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 534 534 of 569 selected Linear Equations In 2 Variables H

  • The line slants sharply down from left to right.
  • The line passes through the following points:
    • (0 comma 6)
    • (1 comma 3)

The graph shows a linear relationship between x and y . Which equation represents this relationship, where R is a positive constant?

  1. Rx+18y=36

  2. Rx-18y=-36

  3. 18x+Ry=36

  4. 18x-Ry=-36

Show Answer Correct Answer: C

Choice C is correct. The equation representing the linear relationship shown can be written in slope-intercept form y = m x + b , where m is the slope and (0,b) is the y-intercept of the line. The line shown passes through the points (0,6) and (2,0). Given two points on a line, (x1,y1) and (x2,y2), the slope of the line can be calculated using the equation m=y2-y1x2-x1. Substituting (0,6) and (2,0) for (x1,y1) and (x2,y2), respectively, in this equation yields m=0-62-0, which is equivalent to m=-62, or m = - 3 . Since (0,6) is the y-intercept, it follows that b = 6 . Substituting - 3 for m and 6 for b in the equation y = m x + b yields y = - 3 x + 6 . Adding 3 x to both sides of this equation yields 3 x + y = 6 . Multiplying this equation by 6 yields 18 x + 6 y = 36 . It follows that the equation 18x+Ry=36, where R is a positive constant, represents this relationship.

Choice A is incorrect. The graph of this relationship passes through the point (0,2), not (0,6).

Choice B is incorrect. The graph of this relationship passes through the point (0,2), not (0,6).

Choice D is incorrect. The graph of this relationship passes through the point (-2,0), not (2,0).

Question 535 535 of 569 selected Linear Equations In 1 Variable E

What value of p satisfies the equation 2 p + 275 = 325 ?

  1. 5

  2. 25

  3. 48

  4. 300

Show Answer Correct Answer: B

Choice B is correct. Subtracting 275 from both sides of the given equation yields 2 p = 50 . Dividing both sides of this equation by 2 yields p = 25 . Therefore, the value of p that satisfies the given equation is 25 .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect. This is the value of p that satisfies the equation (2+p)+275=325, not 2p+275=325.

Choice D is incorrect. This is the value of p that satisfies the equation 2p-275=325, not 2p+275=325.

Question 536 536 of 569 selected Linear Functions E

If y = 5 x + 10 , what is the value of y when x = 8 ?

Show Answer Correct Answer: 50

The correct answer is 50 . Substituting 8 for x in the given equation yields y=5(8)+10, or y=50. Therefore, the value of y is 50 when x=8.

Question 537 537 of 569 selected Linear Functions H

The cost of renting a carpet cleaner is $52 for the first day and $26 for each additional day. Which of the following functions gives the cost C(d), in dollars, of renting the carpet cleaner for d days, where d is a positive integer?

  1. C(d)=26d+26

  2. C(d)=26d+52

  3. C(d)=52d-26

  4. C(d)=52d+78

Show Answer Correct Answer: A

Choice A is correct. It's given that the cost of renting a carpet cleaner is $52 for the first day and $26 for each additional day. Therefore, the cost C(d), in dollars, of renting the carpet cleaner for d days is the sum of the cost for the first day, $52, and the cost for the additional d - 1 days, $26(d-1). It follows that C(d)=52+26(d-1), which is equivalent to C(d)=52+26d-26, or C(d)=26d+26.

Choice B is incorrect. This function gives the cost of renting a carpet cleaner for d days if the cost is $78, not $52, for the first day and $26 for each additional day.

Choice C is incorrect. This function gives the cost of renting a carpet cleaner for d days if the cost is $26, not $52, for the first day and $52, not $26, for each additional day.

Choice D is incorrect. This function gives the cost of renting a carpet cleaner for d days if the cost is $130, not $52, for the first day and $52, not $26, for each additional day.

Question 538 538 of 569 selected Linear Equations In 2 Variables E

Naomi bought both rabbit snails and nerite snails for a total of $52. Each rabbit snail costs $8 and each nerite snail costs $6. If Naomi bought 2 nerite snails, how many rabbit snails did she buy?

  1. 5

  2. 12

  3. 14

  4. 50

Show Answer Correct Answer: A

Choice A is correct. Let x represent the number of rabbit snails that Naomi bought. It’s given that each rabbit snail costs $8. Therefore, the total cost, in dollars, of the rabbit snails that Naomi bought can be represented by the expression 8 x . It’s also given that each nerite snail costs $6, and that Naomi bought 2 nerite snails. Therefore, the total cost, in dollars, of the nerite snails that Naomi bought is 6(2), or 12 . Since Naomi bought both the rabbit snails and the nerite snails for a total of $52, the equation 8 x + 12 = 52 can be used to represent the situation. Subtracting 12 from both sides of this equation yields 8 x = 40 . Dividing both sides of this equation by 8 yields x = 5 . Therefore, Naomi bought 5 rabbit snails.

Choice B is incorrect. This is the total cost, in dollars, of the nerite snails that Naomi bought, not the number of rabbit snails.

Choice C is incorrect. This is the cost, in dollars, of one rabbit snail and one nerite snail, not the number of rabbit snails that Naomi bought.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 539 539 of 569 selected Linear Inequalities In 1 Or 2 Variables M

  • The boundary of the inequality is a dashed line.
    • The line slants sharply down from left to right.
    • The line passes through the following points:
      • (negative 1 comma 5)
      • (0 comma 1)
      • (1 comma negative 3)
  • The area above and to the right of the boundary is shaded.

The shaded region shown represents the solutions to which inequality?

  1. y<1+4x

  2. y<1-4x

  3. y>1+4x

  4. y>1-4x

Show Answer Correct Answer: D

Choice D is correct. The equation for the line representing the boundary of the shaded region can be written in slope-intercept form y=b+mx, where m is the slope and (0,b) is the y-intercept of the line. For the graph shown, the boundary line passes through the points (0,1) and (1,-3). Given two points on a line, (x1,y1) and (x2,y2), the slope of the line can be calculated using the equation m=y2-y1x2-x1. Substituting the points (0,1) and (1,-3) for (x1,y1) and (x2,y2) in this equation yields m=-3-11-0, which is equivalent to m=-41, or m=-4. Since the point (0,1) represents the y-intercept, it follows that b = 1 . Substituting -4 for m and 1 for b in the equation y=b+mx yields y=1-4x as the equation of the boundary line. Since the shaded region represents all the points above this boundary line, it follows that the shaded region shown represents the solutions to the inequality y>1-4x.

Choice A is incorrect. This inequality represents a region below, not above, a boundary line with a slope of 4 , not -4 .

Choice B is incorrect. This inequality represents a region below, not above, the boundary line shown.

Choice C is incorrect. This inequality represents a region whose boundary line has a slope of 4 , not -4 .

Question 540 540 of 569 selected Linear Equations In 2 Variables M

In the xy-plane, line s passes through the point (0,0) and is parallel to the line represented by the equation y = 18 x + 2 . If line s also passes through the point (4,d), what is the value of d ?

  1. 2

  2. 18

  3. 72

  4. 74

Show Answer Correct Answer: C

Choice C is correct. A line in the xy-plane can be represented by an equation of the form y=mx+b, where m is the slope and b is the y-coordinate of the y-intercept of the line. It's given that line s passes through the point (0,0). Therefore, the y-coordinate of the y-intercept of line s is 0 . It's also given that line s is parallel to the line represented by the equation y=18x+2. Since parallel lines have the same slope, it follows that the slope of line s is 18 . Therefore, line s can be represented by the equation y=mx+b, where m = 18 and b = 0 . Substituting 18 for m and 0 for b in y=mx+b yields the equation y=18x+0, or y=18x. If line s passes through the point (4,d), then when x = 4 , y = d for the equation y=18x. Substituting 4 for x and d for y in this equation yields d=18(4), or d = 72 .

Choice A is incorrect. This is the y-coordinate of the y-intercept of the line represented by the equation y=18x+2.

Choice B is incorrect. This is the slope of the line represented by the equation y=18x+2.

Choice D is incorrect. The line represented by the equation y=18x+2, not line s , passes through the point (4,74).

Question 541 541 of 569 selected Linear Equations In 1 Variable E

If 8 x = 6 , what is the value of 72 x ?

  1. 3

  2. 15

  3. 54

  4. 57

Show Answer Correct Answer: C

Choice C is correct. It’s given that 8x=6. Multiplying each side of this equation by 9 yields 72x=54. Therefore, the value of 72 x is 54 .

Choice A is incorrect. This is the value of 4 x , not 72 x .

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 542 542 of 569 selected Systems Of 2 Linear Equations In 2 Variables M

y = 6 x + 16

- 7 x - y = 36

What is the solution (x,y) to the given system of equations?

  1. (-4,-8)

  2. (-2013,-8013)

  3. (4,40)

  4. (20,136)

Show Answer Correct Answer: A

Choice A is correct. The given system of linear equations can be solved by the substitution method. The first equation in the given system of equations defines y as 6x+16. Substituting 6x+16 for y in the second equation of the given system of equations yields -7x-(6x+16)=36. Applying the distributive property on the left-hand side of this equation yields -7x-6x-16=36, or -13x-16=36. Adding 16 to both sides of this equation yields -13x=52. Dividing both sides of this equation by -13 yields x=-4. Substituting -4 for x in the first equation of the given system of equations, y=6x+16, yields y=6(-4)+16, or y=-8. Therefore, the solution (x,y) to the given system of equations is (-4,-8).

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 543 543 of 569 selected Linear Equations In 2 Variables E

Last week, an interior designer earned a total of $1,258 from consulting for x hours and drawing up plans for y hours. The equation 68x+85y=1,258 represents this situation. Which of the following is the best interpretation of 68 in this context?

  1. The interior designer earned $68 per hour consulting last week.

  2. The interior designer worked 68 hours drawing up plans last week.

  3. The interior designer earned $68 per hour drawing up plans last week.

  4. The interior designer worked 68 hours consulting last week.

Show Answer Correct Answer: A

Choice A is correct. It's given that 68x+85y=1,258 represents the situation where an interior designer earned a total of $1,258 last week from consulting for x hours and drawing up plans for y hours. Thus, 68 x represents the amount earned, in dollars, from consulting for x hours, and 85 y represents the amount earned, in dollars, from drawing up plans for y hours. Since 68 x represents the amount earned, in dollars, from consulting for x hours, it follows that the interior designer earned $68 per hour consulting last week.

Choice B is incorrect. The interior designer worked y hours, not 68 hours, drawing up plans last week.

Choice C is incorrect. The interior designer earned $85 per hour, not $68 per hour, drawing up plans last week.

Choice D is incorrect. The interior designer worked x hours, not 68 hours, consulting last week.

Question 544 544 of 569 selected Systems Of 2 Linear Equations In 2 Variables M

x + 3 y = 29

3 y = 11

The solution to the given system of equations is (x,y). What is the value of x ?

Show Answer Correct Answer: 18

The correct answer is 18 . It's given by the second equation in the system that 3 y = 11 . Substituting 11 for 3 y in the first equation in the system, x + 3 y = 29 , yields x + 11 = 29 . Subtracting 11 from both sides of this equation yields x = 18 .

Question 545 545 of 569 selected Systems Of 2 Linear Equations In 2 Variables E

Angela is playing a video game. In this game, players can score points only by collecting coins and stars. Each coin is worth c points, and each star is worth s points.

  • The first time she played, Angela scored 700 points. She collected 20 coins and 10 stars.
  • The second time she played, Angela scored 850 points. She collected 25 coins and 12 stars.

Which system of equations can be used to correctly determine the values of c and s ?

  1. This answer choice consists of two equations. 10 c, plus 20 s, equals 700, and, 12 c, plus 25 s, equals 850

  2. This answer choice consists of two equations. 20 c, plus 10 s, equals 700, and, 25 c, plus 12 s, equals 850

  3. This answer choice consists of two equations. 20 c, plus 700 s, equals 10, and, 25 c, plus 850 s, equals 12

  4. This answer choice consists of two equations. 700 c, plus 20 s, equals 10, and, 850 c, plus 25 s, equals 12

Show Answer Correct Answer: B

Choice B is correct. The number of coins collected can be multiplied by c to give the score from the points earned from coins. Similarly, the number of stars collected can be multiplied by s to give the score from the points earned from the stars. Therefore, the total score each time Angela played is 20 c, plus 10 s, equals 700, and the total score the second time she played is 25 c, plus 12 s, equals 850.

Choices A, C, and D are incorrect and may result from misidentifying the terms of the equation. Choice A switches coins and stars, choice C switches stars and points, and choice D misidentifies coins, stars, and points.

 

Question 546 546 of 569 selected Linear Functions M

The figure presents a 2-column table, with two rows of data, titled “Population of Greenleaf, Idaho.” The heading of the first column is “Year.” The heading of the second column is “Population.” The 2 rows of data are as follows. Row 1. Year, 2000. Population, 862. Row 2. Year, 2010. Population, 846

The table above shows the population of Greenleaf, Idaho, for the years 2000 and 2010. If the relationship between population and year is linear, which of the following functions P models the population of Greenleaf t years after 2000?

 

  1. P of t equals, 862 minus 1 point 6 times t

  2. P of t equals, 862 minus 16 times t

  3. P of t equals, 862 plus 16 times, open parenthesis, t minus 2,000, close parenthesis

  4. P of t equals, 862 minus 1 point 6 times, open parenthesis, t minus 2,000, close parenthesis

Show Answer Correct Answer: A

Choice A is correct. It is given that the relationship between population and year is linear; therefore, the function that models the population t years after 2000 is of the form P of t equals, m, t plus b, where m is the slope and b is the population when t equals 0. In the year 2000, t equals 0. Therefore, b equals 862. The slope is given by m equals, the fraction with numerator P of 10, minus P of 0, and denominator 10 minus 0, end fraction, which equals the fraction with numerator 846 minus 862, and denominator 10 minus 0, end fraction, which equals negative 16 over 10, which equals negative 1 point 6. Therefore, P of t equals, negative 1 point 6, t, plus 862, which is equivalent to the equation in choice A.

Choice B is incorrect and may be the result of incorrectly calculating the slope as just the change in the value of P. Choice C is incorrect and may be the result of the same error as in choice B, in addition to incorrectly using t to represent the year, instead of the number of years after 2000. Choice D is incorrect and may be the result of incorrectly using t to represent the year instead of the number of years after 2000.

 

Question 547 547 of 569 selected Linear Functions M

A linear function f gives a company’s profit, in dollars, for selling x items. The company’s profit is $220 when it sells 8 items, and its profit is $320 when it sells 10 items. Which equation defines f ?

  1. f(x)=150x-320

  2. f(x)=32x

  3. f(x)=50x-10x

  4. f(x)=50x-180

Show Answer Correct Answer: D

Choice D is correct. It’s given that the relationship between x and f(x) is linear. A linear function can be written in the form f(x)=mx+b, where m is the slope and b is the y-coordinate of the y-intercept of the graph of y=f(x) in the xy-plane. Given two points on a line, (x1,y1) and (x2,y2), the slope of the line can be found using the slope formula m=y2-y1x2-x1. It’s given that the company’s profit is $220 when it sells 8 items and the profit is $320 when it sells 10 items. Since f(x) represents the company's profit, in dollars, for selling x items, the graph of y=f(x) in the xy-plane passes through the points (8,220) and (10,320). Substituting (8,220) and (10,320) for (x1,y1) and (x2,y2), respectively, in the slope formula yields m=320-22010-8, which gives m=1002, or m = 50 . Substituting 50 for m , 8 for x , and 220 for f(x) in f(x)=mx+b yields 220=(50)(8)+b, or 220=400+b. Subtracting 400 from each side of this equation yields -180=b. Substituting 50 for m and - 180 for b in f(x)=mx+b yields f(x)=50x-180. Therefore, the equation that defines f is f(x)=50x-180.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Question 548 548 of 569 selected Linear Equations In 1 Variable M

If 9(4-3x)+2=8(4-3x)+18, what is the value of 4-3x?

  1. -16

  2. -4

  3. 4

  4. 16

Show Answer Correct Answer: D

Choice D is correct. The value of 4-3x can be found by isolating this expression in the given equation. Subtracting 2 from both sides of the given equation yields 9(4-3x)=8(4-3x)+16. Subtracting 8(4-3x) from both sides of this equation yields 9(4-3x)-8(4-3x)=16, which gives 1(4-3x)=16, or 4-3x=16. Therefore, the value of 4-3x is 16 .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect. This is the value of x , not 4-3x.

Choice C is incorrect and may result from conceptual or calculation errors.

Question 549 549 of 569 selected Linear Equations In 2 Variables E

The y-intercept of the graph of y = - 6 x - 32 in the xy-plane is (0,y). What is the value of y ?

Show Answer Correct Answer: -32

The correct answer is -32. It’s given that the y-intercept of the graph of y=-6x-32 is (0,y). Substituting 0 for x in this equation yields y=-6(0)-32, or y=-32. Therefore, the value of y that corresponds to the y-intercept of the graph of y=-6x-32 in the xy-plane is -32.

Question 550 550 of 569 selected Linear Equations In 1 Variable E

If 5 x = 20 , what is the value of 15 x ?

  1. 7

  2. 12

  3. 23

  4. 60

Show Answer Correct Answer: D

Choice D is correct. It’s given that 5x=20. Multiplying both sides of this equation by 3 yields 15x=60. Therefore, the value of 15x is 60 .

Choice A is incorrect and may result from conceptual errors.

Choice B is incorrect and may result from conceptual errors.

Choice C is incorrect and may result from conceptual errors.

Question 551 551 of 569 selected Linear Equations In 2 Variables E
The figure presents a graph of a line in the x y-plane with the origin labeled O. The numbers negative five through positive five are indicated along the x- and y-axes.  The line begins in the second quadrant and moves downward and to the right. It crosses the y-axis at 3 and the x-axis at one.

What is the equation of the line shown in the xy-plane above?

  1. y equals 3 x minus 3
  2. y equals negative 3 x plus 3
  3. y equals one-third x minus 3
  4. y equals negative one-third x plus 3
Show Answer Correct Answer: B

Choice B is correct. Any line in the xy-plane can be defined by an equation in the form y = mx + b, where m is the slope of the line and b is the y-coordinate of the y-intercept of the line. From the graph, the y-intercept of the line is (0, 3). Therefore, b = 3. The slope of the line is the change in the value of y divided by the change in the value of x for any two points on the line. The line shown passes through (0, 3) and (1, 0), so m equals, the fraction with numerator 3 minus zero, and denominator zero minus 1, or m = –3. Therefore, the equation of the line is y = –3x + 3.

Choices A and C are incorrect because the equations given in these choices represent a line with a positive slope. However, the line shown has a negative slope. Choice D is incorrect because the equation given in this choice represents a line with slope of negative one third. However, the line shown has a slope of  –3.

Question 552 552 of 569 selected Linear Equations In 1 Variable E

What is the solution to the equation 2 x plus 3, equals 7?

  1. 1

  2. 1.5

  3. 2

  4. 4

Show Answer Correct Answer: C

Choice C is correct. Subtracting 3 from both sides of the given equation yields 2 x equals 4. Dividing both sides by 2 results in x equals 2.

Choices A and B are incorrect and may result from computational errors. Choice D is incorrect. This is the value of 2 x.

Question 553 553 of 569 selected Linear Functions E

f(x)=x+b

For the linear function f , b is a constant. When x=0, f(x)=30. What is the value of b ?

  1. -30

  2. - 1 30

  3. 1 30

  4. 30

Show Answer Correct Answer: D

Choice D is correct. It’s given that when x = 0 , f(x)=30. Substituting 0 for x and 30 for f(x) in the given function yields 30=0+b, or 30 = b . Therefore, the value of b is 30 .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Question 554 554 of 569 selected Systems Of 2 Linear Equations In 2 Variables H

  • For the first line in the system:
    • The line slants gradually down from left to right.
    • The line passes through the following points:
      • (negative 4 comma 5)
      • (0 comma 4)
      • (8 comma 2)
  • For the second line in the system:
    • The line is horizontal.
    • The line passes through the following points:
      • (0 comma 2)
      • (8 comma 2)

If a new graph of three linear equations is created using the system of equations shown and the equation x + 4 y = -16 , how many solutions (x,y) will the resulting system of three equations have?

  1. Zero

  2. Exactly one

  3. Exactly two

  4. Infinitely many

Show Answer Correct Answer: A

Choice A is correct. A solution to a system of equations must satisfy each equation in the system. It follows that if an ordered pair (x,y) is a solution to the system, the point (x,y) lies on the graph in the xy-plane of each equation in the system. The only point that lies on each graph of the system of two linear equations shown is their intersection point (8,2).  It follows that if a new graph of three linear equations is created using the system of equations shown and the graph of x+4y=-16, this system has either zero solutions or one solution, the point (8,2). Substituting 8 for x and 2 for y in the equation x+4y=-16 yields 8+4(2)=-16, or 16=-16. Since this equation is not true, the point (8,2) does not lie on the graph of x+4y=-16. Therefore, (8,2) is not a solution to the system of three equations. It follows that there are zero solutions to this system.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 555 555 of 569 selected Linear Equations In 2 Variables E

Davio bought some potatoes and celery. The potatoes cost $0.69 per pound, and the celery cost $0.99 per pound. If Davio spent $5.34 in total and bought twice as many pounds of celery as pounds of potatoes, how many pounds of celery did Davio buy?

  1. 2

  2. 2.5

  3. 2.67

  4. 4

Show Answer Correct Answer: D

Choice D is correct. Let p represent the number of pounds of potatoes and let c represent the number of pounds of celery that Davio bought. It’s given that potatoes cost $0.69 per pound and celery costs $0.99 per pound. If Davio spent $5.34 in total, then the equation 0.69p+0.99c=5.34 represents this situation. It’s also given that Davio bought twice as many pounds of celery as pounds of potatoes; therefore, c=2p. Substituting 2p for c in the equation 0.69p+0.99c=5.34 yields 0.69p+0.99(2p)=5.34, which is equivalent to 0.69p+1.98p=5.34, or 2.67p=5.34. Dividing both sides of this equation by 2.67 yields p=2. Substituting 2 for p in the equation c=2p yields c=2(2), or c=4. Therefore, Davio bought 4 pounds of celery.

Choice A is incorrect. This is the number of pounds of potatoes, not the number of pounds of celery, Davio bought.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Question 556 556 of 569 selected Linear Functions E

The function g is defined by g(x)=10x+8. What is the value of g(x) when x = 8 ?

  1. 0

  2. 8

  3. 10

  4. 88

Show Answer Correct Answer: D

Choice D is correct. The value of g(x) when x=8 can be found by substituting 8 for x in the given equation g(x)=10x+8. This yields g(8)=10(8)+8, or g(8)=88. Therefore, when x=8, the value of g(x) is 88 .

Choice A is incorrect. This is the value of x when g(x)=8, rather than the value of g(x) when x=8.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Question 557 557 of 569 selected Linear Equations In 1 Variable E

If x = 40 , what is the value of x + 6 ?

  1. 34

  2. 40

  3. 46

  4. 64

Show Answer Correct Answer: C

Choice C is correct. It's given that x=40. Adding 6 to both sides of this equation yields x+6=40+6, or x+6=46. Therefore, the value of x+6 is 46 .

Choice A is incorrect. This is the value of x-6, not x+6.

Choice B is incorrect. This is the value of x , not x+6.

Choice D is incorrect. This is the value of x+24, not x+6.
 

Question 558 558 of 569 selected Linear Equations In 1 Variable M

A candle is made of 17 ounces of wax. When the candle is burning, the amount of wax in the candle decreases by 1 ounce every 4 hours. If 6 ounces of wax remain in this candle, for how many hours has it been burning?

  1. 3

  2. 6

  3. 24

  4. 44

Show Answer Correct Answer: D

Choice D is correct. It’s given that the candle starts with 17 ounces of wax and has 6 ounces of wax remaining after a period of time has passed. The amount of wax the candle has lost during the time period can be found by subtracting the remaining amount of wax from the amount of wax the candle was made of, which yields 17-6 ounces, or 11 ounces. This means the candle loses 11 ounces of wax during that period of time. It’s given that the amount of wax decreases by 1 ounce every 4 hours. If h represents the number of hours the candle has been burning, it follows that 14=11h. Multiplying both sides of this equation by 4 h yields h = 44 . Therefore, the candle has been burning for 44 hours. 

Choice A is incorrect and may result from using the equation 14=h11 rather than 14=11h to represent the situation, and then rounding to the nearest whole number. 

Choice B is incorrect. This is the amount of wax, in ounces, remaining in the candle, not the number of hours it has been burning. 

Choice C is incorrect and may result from using the equation 14=6h rather than 14=11h to represent the situation. 

Question 559 559 of 569 selected Systems Of 2 Linear Equations In 2 Variables E

Equation 1: 2 x, plus 7 y, equals 9. Equation 2: 8 x, plus 28 y, equals a

In the given system of equations, a is a constant. If the system has infinitely many solutions, what is the value of a ?

  1. 4

  2. 9

  3. 36

  4. 54

Show Answer Correct Answer: C

Choice C is correct. A system of two linear equations has infinitely many solutions if one equation is equivalent to the other. This means that when the two equations are written in the same form, each coefficient or constant in one equation is equal to the corresponding coefficient or constant in the other equation multiplied by the same number. The equations in the given system of equations are written in the same form, with x and y on the left-hand side of the equation and a constant on the right-hand side of the equation. The coefficients of x and y in the second equation are equal to the coefficients of x and y, respectively, in the first equation multiplied by 4: 8 equals, 2 times 4 and 28 equals, 7 times 4. Therefore, the constant in the second equation must be equal to 4 times the constant in the first equation: a, equals, 9 times 4, or a, equals 36.

Choices A, B, and D are incorrect. When a, equals 4, a, equals 9, or a, equals 54, the given system of equations has no solution.

 

Question 560 560 of 569 selected Linear Functions E

For the linear function f, the graph of y=f(x) in the xy-plane has a slope of 14 and passes through the point (0,5). Which equation defines f?

  1. f(x)=14x+5

  2. f(x)=14x+15

  3. f(x)=14x-54

  4. f(x)=14x-5

Show Answer Correct Answer: A

Choice A is correct. An equation that defines a linear function f can be written in the form f(x)=mx+b, where m is the slope of the graph of y=f(x) in the xy-plane and (0,b) is the y-intercept of the graph. It’s given that for the linear function f, the graph of y=f(x) in the xy-plane has a slope of 14. Therefore, m=14. It’s also given that the graph of y=f(x) passes through the point (0,5). Therefore, the y-intercept of the graph is (0,5), and it follows that b=5. Substituting 14 for m and 5 for b in the equation f(x)=mx+b yields f(x)=14x+5.

Choice B is incorrect. This equation defines a function whose graph has a y-intercept of (0,15), not (0,5).

Choice C is incorrect. This equation defines a function whose graph has a y-intercept of (0,-54), not (0,5).

Choice D is incorrect. This equation defines a function whose graph has a y-intercept of (0,-5), not (0,5).

Question 561 561 of 569 selected Linear Inequalities In 1 Or 2 Variables E

Monarch butterflies can fly only with a body temperature of at least 55.0 degrees Fahrenheit (°F). If a monarch butterfly's body temperature is 51.3 °F, what is the minimum increase needed in its body temperature, in °F, so that it can fly?

  1. 1.3

  2. 3.7

  3. 5.0

  4. 6.3

Show Answer Correct Answer: B

Choice B is correct. It's given that monarch butterflies can fly only with a body temperature of at least 55.0 degrees Fahrenheit (°F). Let x represent the minimum increase needed in the monarch butterfly's body temperature to fly. If the monarch butterfly's body temperature is 51.3°F, the inequality 51.3+x55.0 represents this situation. Subtracting 51.3 from both sides of this inequality yields x3.7. Therefore, if the monarch butterfly's body temperature is 51.3°F, the minimum increase needed in its body temperature, in °F, so that it can fly is 3.7.

Choice A is incorrect. This is the minimum increase needed in body temperature if the monarch butterfly's body temperature is 53.7°F, not 51.3°F.

Choice C is incorrect. This is the minimum increase needed in body temperature if the monarch butterfly's body temperature is 50.0°F, not 51.3°F.

Choice D is incorrect. This is the minimum increase needed in body temperature if the monarch butterfly's body temperature is 48.7°F, not 51.3°F.

Question 562 562 of 569 selected Linear Equations In 2 Variables H
x y
-18 -48
7 52

The table shows two values of x and their corresponding values of y . In the xy-plane, the graph of the linear equation representing this relationship passes through the point (17, a). What is the value of a ?

  1. - 4 11

  2. - 4 77

  3. 4 7

  4. 172 7

Show Answer Correct Answer: D

Choice D is correct. The linear relationship between x and y can be represented by the equation y=mx+b, where m is the slope of the graph of this equation in the xy-plane and b is the y-coordinate of the y-intercept. The slope of a line between any two points (x1,y1) and (x2,y2) on the line can be calculated using the slope formula m=y2-y1x2-x1. Based on the table, the graph contains the points (-18,-48) and (7,52). Substituting (-18,-48) and (7,52) for (x1,y1) and (x2,y2), respectively, in the slope formula yields m=52-(-48)7-(-18), which is equivalent to m=10025, or m=4. Substituting 4 for m , - 18 for x , and - 48 for y in the equation y=mx+b yields -48=4(-18)+b, or -48=-72+b. Adding 72 to both sides of this equation yields 24=b. Therefore, m=4 and b=24. Substituting 4 for m and 24 for b in the equation y=mx+b yields y=4x+24. Thus, the equation y=4x+24 represents the linear relationship between x and y . It's also given that the graph of the linear equation representing this relationship in the xy-plane passes through the point (17,a). Substituting 17 for x and a for y in the equation y=4x+24 yields a=4(17)+24, which is equivalent to a=47+1687, or a=1727.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Question 563 563 of 569 selected Linear Inequalities In 1 Or 2 Variables E

A cleaning service that cleans both offices and homes can clean at most 14 places per day. Which inequality represents this situation, where f is the number of offices and h is the number of homes?

  1. f+h14

  2. f+h14

  3. f-h14

  4. f-h14

Show Answer Correct Answer: A

Choice A is correct. It's given that the cleaning service cleans both offices and homes, where f is the number of offices and h is the number of homes the cleaning service can clean per day. Therefore, the expression f + h represents the number of places the cleaning service can clean per day. It's also given that the cleaning service can clean at most 14 places per day. Since f + h represents the number of places the cleaning service can clean per day and the service can clean at most 14 places per day, it follows that the inequality f+h14 represents this situation.

Choice B is incorrect. This inequality represents a cleaning service that cleans at least 14 places per day.

Choice C is incorrect. This inequality represents a cleaning service that cleans at most 14 more offices than homes per day.

Choice D is incorrect. This inequality represents a cleaning service that cleans at least 14 more offices than homes per day.

Question 564 564 of 569 selected Linear Equations In 2 Variables M

A store sells two different-sized containers of blueberries. The store’s sales of these blueberries totaled 896.86 dollars last month. The equation 4.51 x + 6.07 y = 896.86 represents this situation, where x is the number of smaller containers sold and y is the number of larger containers sold. According to the equation, what is the price, in dollars, of each smaller container?

Show Answer Correct Answer: 4.51, 451/100

The correct answer is 4.51 . It’s given that the equation 4.51 x + 6.07 y = 896.86 represents this situation, where x is the number of smaller containers sold, y is the number of larger containers sold, and 896.86 is the store’s total sales, in dollars, of blueberries last month. Therefore, 4.51 x represents the store's sales, in dollars, of smaller containers, and 6.07 y represents the store's sales, in dollars, of larger containers. Since x is the number of smaller containers sold, the price, in dollars, of each smaller container is 4.51 .

Question 565 565 of 569 selected Linear Functions E

f(x)=7x+1

The function gives the total number of people on a company retreat with x managers. What is the total number of people on a company retreat with 7 managers?

Show Answer Correct Answer: 50

The correct answer is 50 . It's given that the function f gives the total number of people on a company retreat with x managers. It's also given that 7 managers are on the company retreat. Substituting 7 for x in the given function yields f(7)=7(7)+1, or f(7)=50. Therefore, there are a total of 50 people on a company retreat with 7 managers.

Question 566 566 of 569 selected Linear Functions E

Which of the following is the graph of the equation y, equals 2 x minus 5 in the xy-plane?

  1.  

    The figure presents the graph of a line in the x y-plane with the origin labeled O. The numbers negative five through five are indicated on each axis. The line begins in quadrant 3 and moves upward and to the right. It crosses the x-axis at negative 5 and the y-axis at 2 point 5, ending in quadrant 1

     

  2.  

    The figure presents the graph of a line in the x y-plane with the origin labeled O. The numbers negative five through five are indicated on each axis. The line begins in quadrant 2 and moves downward and to the right. It crosses the x-axis at negative 2 point 5 and the y-axis at negative 5, ending in quadrant 4

     

  3.  

    The figure presents the graph of a line in the x y-plane with the origin labeled O. The numbers negative five through five are indicated on each axis. The line begins in quadrant 3 and moves upward and to the right. It crosses the x-axis at negative 2 point 5 and the y-axis at 5, ending in quadrant 1

     

  4.  

    The figure presents the graph of a line in the x y-plane with the origin labeled O. The numbers negative five through five are indicated on each axis. The line begins in quadrant 3 and moves upward and to the right. It crosses the y-axis at negative 5 and the x-axis at 2 point 5, ending in quadrant 1

     

Show Answer Correct Answer: D

Choice D is correct. In the xy-plane, the graph of the equation y equals, m x plus b, where m and b are constants, is a line with slope m and y-intercept with coordinates 0 comma b. Therefore, the graph of y equals, 2 x minus 5 in the xy-plane is a line with slope 2 and a y-intercept with coordinates 0 comma negative 5. Having a slope of 2 means that for each increase in x by 1, the value of y increases by 2. Only the graph in choice D has a slope of 2 and crosses the y-axis at the point with coordinates 0 comma negative 5. Therefore, the graph shown in choice D must be the correct answer.

Choices A, B, and C are incorrect. The graph of y equals, 2 x minus 5 in the xy-plane is a line with slope 2 and a y-intercept at the point with coordinates 0 comma negative 5. The graph in choice A crosses the y-axis at the point with coordinates 0 comma 2 point 5, not the point with coordinates 0 comma negative 5, and it has a slope of one-half, not 2. The graph in choice B crosses the y-axis at the point with coordinates 0 comma negative 5; however, the slope of this line is negative 2, not 2. The graph in choice C has a slope of 2; however, the graph crosses the y-axis at the point with coordinates 0 comma 5, not the point with coordinates 0 comma negative 5.

 

Question 567 567 of 569 selected Linear Inequalities In 1 Or 2 Variables M
y is less than or equal to, 3 x plus 1
x minus y is greater than 1

Which of the following ordered pairs (x, y) satisfies the system of inequalities above?

  1. negative 2 comma negative 1​​​​​​​

  2. negative 1 comma 3​​​​​​​

  3. 1 comma 5​​​​​​​

  4. 2 comma negative 1​​​​​​​

Show Answer Correct Answer: D

Choice D is correct. Any point (x, y) that is a solution to the given system of inequalities must satisfy both inequalities in the system. The second inequality in the system can be rewritten as x is greater than, y plus 1. Of the given answer choices, only choice D satisfies this inequality, because inequality 2 is greater than, negative 1 plus 1 is a true statement. The point with coordinates 2 comma negative 1 also satisfies the first inequality.

Alternate approach: Substituting the ordered pair 2 comma negative 1 into the first inequality gives negative 1 is less than or equal to, 3 times 2, plus 1, or negative 1 is less than or equal to 7, which is a true statement. Substituting the ordered pair 2 comma negative 1 into the second inequality gives 2 minus negative 1, is greater than 1, or 3 is greater than 1, which is a true statement. Therefore, since the ordered pair 2 comma negative 1 satisfies both inequalities, it is a solution to the system.

Choice A is incorrect because substituting −2 for x and −1 for y in the first inequality gives negative 1 is less than or equal to, 3 times negative 2, plus 1, or negative 1 is less than or equal to negative 5, which is false. Choice B is incorrect because substituting −1 for x and 3 for y in the first inequality gives 3 is less than or equal to, 3 times negative 1, plus 1, or 3 is less than or equal to negative 2, which is false. Choice C is incorrect because substituting 1 for x and 5 for y in the first inequality gives 5 is less than or equal to, 3 times 1, plus 1, or 5 is less than or equal to 4, which is false.

 

Question 568 568 of 569 selected Linear Equations In 2 Variables M

20406080100x1020304050yONumber of T-shirtsNumber of sweatshirts
  • The line slants gradually down from left to right.
  • The line passes through the following points:
    • (0 comma 35)
    • (90 comma 0)

The graph models the relationship between the number of T-shirts, x , and the number of sweatshirts, y , that Kira can purchase for a school fundraiser. Which equation could represent this relationship?

  1. y = 7 x + 18

  2. 7 x + 18 y = 630

  3. y = 18 x + 7

  4. 18 x + 7 y = 630

Show Answer Correct Answer: B

Choice B is correct. A line in the xy-plane can be written as y=mx+b, where m is the slope of the line and b is the y-coordinate of the y-intercept. The graph shown is a line passing through the points (0,35) and (90,0). Substituting 0 for x and 35 for y in the equation y=mx+b yields 35=m(0)+b, or 35=b. Substituting 35 for b, 90 for x, and 0 for y in the equation y=mx+b yields 0=90m+35. Subtracting 35 from both sides of this equation yields -35=90m. Dividing both sides of this equation by 90 yields -3590=m, or -718=m. Substituting -718 for m and 35 for b in the equation y=mx+b yields y=-718x+35. Multiplying both sides of this equation by 18 yields 18y=-7x+35(18), or 18y=-7x+630. Adding 7x to both sides of this equation yields 7x+18y=630. Therefore, the equation 7x+18y=630 represents the relationship between x and y modeled by the graph.

Choice A is incorrect. The point (0,35) is not on the graph of this equation, since 7(0)+18=18, not 35.

Choice C is incorrect. The point (0,35) is not on the graph of this equation, since 18(0)+7=7, not 35.

Choice D is incorrect. The point (90,0) is not on the graph of this equation, since 18(90)+7(0)=1,620, not 630.

Question 569 569 of 569 selected Systems Of 2 Linear Equations In 2 Variables H

5 y = 10 x + 11

- 5 y = 5 x - 21

The solution to the given system of equations is (x,y). What is the value of 30 x ?

Show Answer Correct Answer: 20

The correct answer is 20 . Adding the first equation to the second equation in the given system yields 5y-5y=10x+5x+11-21, or 0=15x-10. Adding 10 to both sides of this equation yields 10=15x. Multiplying both sides of this equation by 2 yields 20=30x. Therefore, the value of 30 x is 20 .